Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 791786, 14 pages
http://dx.doi.org/10.1155/2013/791786
Research Article
Infinite Horizon Optimal Control of Stochastic Delay Evolution
Equations in Hilbert Spaces
Xueping Zhu1 and Jianjun Zhou2
1
2
School of Astronautics, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, China
College of Science, Northwest A&F University, Yangling, Shaanxi 712100, China
Correspondence should be addressed to Jianjun Zhou; zhoujj198310@163.com
Received 6 September 2012; Accepted 16 December 2012
Academic Editor: Juan J. Trujillo
Copyright © 2013 X. Zhu and J. Zhou. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The aim of the present paper is to study an infinite horizon optimal control problem in which the controlled state dynamics
is governed by a stochastic delay evolution equation in Hilbert spaces. The existence and uniqueness of the optimal control
are obtained by means of associated infinite horizon backward stochastic differential equations without assuming the Gâteaux
differentiability of the drift coefficient and the diffusion coefficient. An optimal control problem of stochastic delay partial
differential equations is also given as an example to illustrate our results.
1. Introduction
In this paper, we consider a controlled stochastic evolution
equation of the following form:
𝑑𝑋𝑢 (𝑠) = 𝐴𝑋𝑢 (𝑠) 𝑑𝑠 + 𝐹 (𝑠, 𝑋𝑠𝑢 ) 𝑑𝑠
+ 𝐺 (𝑠, 𝑋𝑠𝑢 ) 𝑅 (𝑠, 𝑋𝑠𝑢 , 𝑢 (𝑠)) 𝑑𝑠 + 𝐺 (𝑠, 𝑋𝑠𝑢 ) 𝑑𝑊 (𝑠) ,
𝑠 ≥ 𝑡,
𝑋𝑡𝑢 = 𝑥,
(1)
where
𝑋𝑠𝑢 (𝑙) = 𝑋𝑢 (𝑠 + 𝑙) ,
𝑙 ∈ [−𝜏, 0] , 𝑥 ∈ 𝐶 ([−𝜏, 0] , 𝐻) . (2)
𝑢 is the control process in a measurable space (𝑈, U),and
𝑊 is a cylindrical Wiener process in a Hilbert space Ξ. 𝐴
is the generator of a strongly continuous semigroup of
bounded linear operator in another Hilbert space 𝐻, and
the coefficients 𝐹 and 𝐺, defined on [0, ∞) × 𝐶 ([−𝜏, 0], 𝐻),
are assumed to satisfy Lipschitz conditions with respect to
appropriate norms. We introduce the cost function
∞
𝐽 (𝑢) = 𝐸 ∫ 𝑒−𝜆𝑠 𝑔 (𝑠, 𝑋𝑠𝑢 , 𝑢 (𝑠)) 𝑑𝑠.
𝑡
(3)
Here, 𝑔 is a given real function, 𝜆 is large enough, and the
control problem is understood in the weak sense. We wish to
minimize the cost function over all admissible controls.
The particular form of the control system is essential
for our results, but it covers numerous interesting cases. For
example, in the particular cases 𝑈 = 𝐻 and 𝑅(𝑡, 𝑥, 𝑢) = 𝑢, the
term 𝑢(𝑡)𝑑𝑡 + 𝑑𝑊(𝑡) in the state equation can be considered
as a control affected by noise.
The stochastic optimal control problem was considered
in 1977 by Bismut [1]. The optimal control problem for
stochastic partial differential equations in the framework of
a compact control state space has been studied in [2–5].
Buckdahn and Raşcanu [6] considered an optimal control
problem for a semilinear parabolic stochastic differential
equation with a nonlinear diffusion coefficient, and the
existence of a quasioptimal (nonrelaxed) control is showed
without assuming convexity of the coefficients. In [7–11], the
authors provided a direct (classical or mild) solution of the
Hamilton-Jacobi-Bellman equation for the value function,
which is then used to prove that the optimal control is
related to the corresponding optimal trajectory by a feedback
law. In Gozzi [10, 11], the existence and uniqueness of a
mild solution of the associated Hamilton-Jacobi-Bellman
equation are proved, when the diffusion term only satisfies
weak nondegeneracy conditions. The proofs are based on
2
the corresponding regularity properties of the transition
semigroup of the associated Ornstein-Uhlenbeck process.
The main tools for the control problem are techniques
from the theory of backward stochastic differential equations
(BSDEs) in the sense of Pardoux and Peng, first considered
in the nonlinear case in [12]; see [13, 14] as general references.
BSDEs have been successfully applied to control problems;
see, for example, [15, 16] and we also refer the reader to
[17–20]. Fuhrman and Tessiture [19] considered the optimal
control problem for stochastic differential equation in the
strong form, assuming Lipschitz conditions and allowing
degeneracy of the diffusion coefficient, under some structural
constraint on the state equation. Existence of an optimal
control for stochastic systems in infinite dimensional spaces
also has been obtained in [21–27]. In [21], Fuhrman and Tessitore showed the regularity with respect to parameters and
the regularity in the Malliavin spaces for the solution of the
backward-forward system and defined the feedback law by
Malliavin calculus. Finally, the optimal control is obtained by
the feedback. Appealing to the Malliavin calculus, compared
with Fuhrman et al. [23], the existence of optimal control
for stochastic differential equations with delay is proved by
the feedback law. Fuhrman and Tessiture [24] dealt with an
infinite horizon optimal control problem for the stochastic evolution equation in Hilbert space, and the optimal
control is showed by means of infinite horizon backward
stochastic differential equation in infinite dimensional spaces
and Malliavin calculus. In Masiero [25], the infinite horizon
optimal control problem for stochastic evolution equation
is also studied by means of the Hamilton-Jacobi-Bellman
equation. In Fuhrman [26], a class of optimal control problems governed by stochastic evolution equations in Hilbert
spaces which includes state constraints is considered, and
the optimal control is obtained by the Fleming logarithmic
transformation. Masiero [27] studied stochastic evolution
equations evolving in a Banach space where 𝐺 is a constant
and characterized the optimal control via a feedback law by
avoiding use of Malliavin calculus. Since there is a lack of
regularity of 𝐹 and 𝐺, Malliavin calculus is not available in
this case; the method in [27] also cannot be used as 𝐺 is
not a constant, but we can prove a theorem similar to [26,
Proposition 3.2], which will be used to define the feedback
law.
In the present paper, we study the infinite horizon optimal
control problem for stochastic delay evolution equations in
Hilbert spaces, and by using Theorem 10, the optimal control
is obtained. Since we do not relate the optimal feedback law
with the gradient of the value function and do not consider
the associated Hamilton-Jacobi-Bellman equation, we can
drop the Gâteaux differentiability of the drift term and the
diffusion term.
The plan of the paper is as follows. In the next section,
some notations are fixed, and the stochastic delay evolution
equations are considered with an infinite horizon; in particular, continuous dependence on initial value (𝑡, 𝑥) is proved. In
Section 3, we give the proof of Theorem 10, which is the key
of many subsequent results. The addressed optimal control
problem is considered, and the fundamental relation between
the optimal control problem and BSDEs is established in
Abstract and Applied Analysis
Section 4. Section 5 is devoted to proving the existence and
uniqueness of optimal control in the weak sense. Finally, an
application is given in Section 6.
2. Preliminaries
We list some notations that are used in this paper. We use
the symbol | ⋅ | to denote the norm in a Banach space 𝐹,
with a subscript if necessary. Let Ξ, 𝐻, and 𝐾 denote real
separable Hilbert spaces, with scalar products (⋅, ⋅)Ξ , (⋅, ⋅)𝐻,
and (⋅, ⋅)𝐾 , respectively. For fixed 𝜏 > 0, C = 𝐶([−𝜏, 0], 𝐻)
denotes the space of continuous functions from [−𝜏, 0] to 𝐻,
endowed with the usual norm |𝑓|𝐶 = sup𝜃∈[−𝜏,0] |𝑓(𝜃)|𝐻. Let
Ξ∗ denote the dual space of Ξ, with scalar product (⋅, ⋅)Ξ∗ , and
let 𝐿(Ξ, 𝐻) denote the space of all bounded linear operators
from Ξ into 𝐻; the subspace of Hilbert-Schmidt operators,
with the Hilbert-Schmidt norm, is denoted by 𝐿 2 (Ξ, 𝐻).
Let (Ω, F, 𝑃) be a complete space with a filtration {F𝑡 }𝑡≥0
which satisfies the usual condition. By a cylindrical Wiener
process with values in a Hilbert space Ξ, defined on (Ω, F, 𝑃),
we mean a family {𝑊(𝑡), 𝑡 ≥ 0} of linear mappings Ξ →
𝐿2 (Ω) such that for every 𝜉, 𝜂 ∈ Ξ, {𝑊(𝑡)𝜉, 𝑡 ≥ 0} is a
real Wiener process and 𝐸(𝑊(𝑡)𝜉 ⋅ 𝑊(𝑡)𝜂) = (𝜉, 𝜂)Ξ . In
the following, {𝑊(𝑡), 𝑡 ≥ 0} is a cylindrical Wiener process
adapted to the filtration {F𝑡 }𝑡≥0 .
In this section and the next section, {F𝑡 }𝑡≥0 will denote
the natural filtration of 𝑊, augmented with the family N
of 𝑃-null of F. The filtration {F𝑡 , 𝑡 ≥ 0} satisfies the usual
conditions. For [𝑎, 𝑏], [𝑎, ∞) ⊂ [0, ∞), we also use the
following notations:
F[𝑎,𝑏] = 𝜎 (𝑊 (𝑠) − 𝑊 (𝑎) : 𝑠 ∈ [𝑎, 𝑏]) ∨ N,
F[𝑎,∞) = 𝜎 (𝑊 (𝑠) − 𝑊 (𝑎) : 𝑠 ∈ [a, ∞)) ∨ N.
(4)
By P we denote the predictable 𝜎-algebra, and by B(Λ) we
denote the Borel 𝜎-algebra of any topological space Λ.
Similar to [24], we define several classes of stochastic
processes with values in a Banach space 𝐹 as follows.
(i) 𝐿2P (Ω × [𝑡, ∞); 𝐹) denotes the space of equivalence
classes of processes 𝑌 ∈ 𝐿2 (Ω × [𝑡, ∞); 𝐹), admitting
a predictable version. 𝐿2P (Ω × [𝑡, ∞); 𝐹) is endowed
with the norm
∞
|𝑌|2 = 𝐸 ∫ |𝑌(𝑠)|2 𝑑𝑠.
(5)
𝑡
𝑝
𝑞
(ii) 𝐿 P (Ω; 𝐿 𝛽 ([𝑡, ∞); 𝐹)), defined for 𝛽 ∈ 𝑅 and 𝑝, 𝑞 ∈
[1, ∞), denotes the space of equivalence classes of
processes {𝑌(𝑠), 𝑠 ≥ 𝑡}, with values in 𝐹, such that the
norm
∞
|𝑌|𝑝 = 𝐸(∫ 𝑒𝑞𝛽𝑠 |𝑌 (𝑠)|𝑞 𝑑𝑠)
𝑝/𝑞
(6)
𝑡
is finite and 𝑌 admits a predictable version.
𝑝
𝑝
(iii) K𝛽 (𝑡) denotes the space 𝐿 P (Ω; 𝐿2𝛽 ([𝑡, ∞); 𝐹)) ×
𝑝
𝐿 P (Ω; 𝐿2𝛽 ([𝑡, ∞); 𝐿 2 (Ξ, 𝐹))). The norm of an element
Abstract and Applied Analysis
3
𝑝
(𝑌, 𝑍) ∈ K𝛽 is |(𝑌, 𝑍)| = |𝑌| + |𝑍|. Here, 𝐹 is a Hilbert
space.
𝑒𝑠𝐴 𝐺(𝑡, 𝑥) ∈ 𝐿 2 (Ξ, 𝐻) for every s > 0, 𝑡 ∈ [0, ∞) and 𝑥 ∈ C,
and
𝑝
(iv) 𝐿 P (Ω; 𝐶([𝑡, 𝑇]; 𝐹)), defined for 𝑇 > 𝑡 ≥ 0 and
𝑝 ∈ [1, ∞), denotes the space of predictable processes
{𝑌(𝑠), 𝑠 ∈ [𝑡, 𝑇]} with continuous paths in 𝐹, such that
the norm
|𝑌|𝑝 = 𝐸 sup |𝑌 (𝑠)|𝑝
󵄨
󵄨󵄨 𝑠𝐴
𝜔𝑠 −𝛾
󵄨󵄨𝑒 𝐺 (𝑡, 𝑥)󵄨󵄨󵄨
󵄨𝐿 2 (Ξ,𝐻) ≤ 𝐿𝑒 𝑠 (1 + |𝑥|𝐶) ,
󵄨
󵄨
󵄨󵄨 𝑠𝐴
󵄨󵄨
𝜔𝑠 −𝛾 󵄨󵄨
(12)
󵄨󵄨𝑒 𝐺 (𝑡, 𝑥) − 𝑒𝑠𝐴 𝐺 (𝑡, 𝑦)󵄨󵄨󵄨
󵄨𝐿 2 (Ξ,𝐻) ≤ 𝐿𝑒 𝑠 (󵄨󵄨𝑥 − 𝑦󵄨󵄨𝐶) ,
󵄨
(7)
𝑠∈[𝑡,𝑇]
𝑠 > 0,
𝑡 ∈ [0, +∞) ,
𝑥, 𝑦 ∈ C,
𝑝
is finite. Elements of 𝐿 P (Ω; 𝐶([𝑡, 𝑇]; 𝐹)) are identified
up to indistinguishability.
𝑞
(v) 𝐿 P (Ω; 𝐶𝜂 ([𝑡, ∞); 𝐹)), defined for 𝜂 ∈ 𝑅 and 𝑞 ∈
[1, ∞), denotes the space of predictable processes
{𝑌(𝑠), 𝑠 ≥ 𝑡} with continuous paths in 𝐹, such that the
norm
𝑞
𝜂𝑞𝑠
|𝑌| = 𝐸 sup 𝑒
𝑠≥𝑡
𝑞
|𝑌 (𝑠)|
(8)
for some constants 𝐿 > 0 and 𝛾 ∈ [0, 1/2).
We say that 𝑋 is a mild solution of (10) if it is a continuous,
{F𝑡 }𝑡≥0 -predictable process with values in 𝐻, and it satisfies
𝑃-a.s.,
𝑠
𝑋 (𝑠) = 𝑒(𝑠−𝑡)𝐴 𝑥 (0) + ∫ 𝑒(𝑠−𝜎)𝐴 𝐹 (𝜎, 𝑋𝜎 ) 𝑑𝜎
𝑡
𝑠
+ ∫ 𝑒(𝑠−𝜎)𝐴 𝐺 (𝜎, 𝑋𝜎 ) 𝑑𝑊 (𝜎) ,
𝑞
is finite. Elements of 𝐿 P (Ω; 𝐶𝜂 (𝐹)) are identified up
to indistinguishability.
(vi) Finally, for 𝜂 ∈ 𝑅 and 𝑞 ∈ [1, ∞), we
𝑞
defined Hq𝜂 (𝑡) as the space 𝐿 P (Ω; 𝐿𝑞𝜂 ([𝑡, ∞); 𝐹)) ∩
𝑞
𝐿 P (Ω; 𝐶𝜂 ([𝑡, ∞); 𝐹)), endowed with the norm
|𝑌|H𝑞𝜂 = |𝑌|𝐿𝑞P (Ω;𝐿𝑞𝜂 ([𝑡,∞);𝐹)) + |𝑌|𝐿𝑞P (Ω;𝐶𝜂 ([𝑡,∞);𝐹)) .
𝑝
𝑞
(9)
𝑞
For simplicity, we denote 𝐿 P (Ω; 𝐿 𝛽 ([0, ∞); 𝐹)), 𝐿 P (Ω;
𝑝
𝑝
𝑞
𝑞
𝐶𝜂 ([0, ∞); 𝐹)), H𝑞𝜂 (0), and K𝛽 (0) by 𝐿 P (Ω; 𝐿 𝛽 (𝐹)), 𝐿 P (Ω;
𝑝
𝐶𝜂 (𝐹)), H𝑞𝜂 , and K𝛽 , respectively.
Now, for every fixed 𝑡 ≥ 0, we consider the following
stochastic delay evolution equation:
𝑡
To stress dependence on initial data, we denote the solution
by 𝑋(𝑠, 𝑡, 𝑥). Note that 𝑋(𝑠, 𝑡, 𝑥) is F[𝑡,𝑠] measurable, hence,
independent of F𝑡 .
We first recall a well-known result on solvability of (10)
on bounded interval.
Theorem 1. Assume that Hypothesis 1 holds. Then, for all
𝑞 ∈ [2, ∞) and 𝑇 > 0, there exists a unique process 𝑋 ∈
𝑞
𝐿 P (Ω, 𝐶([𝑡, 𝑇]; 𝐻)) as mild solution of (10). Moreover,
𝑞
𝐸 sup |𝑋 (𝑠)|𝑞 ≤ 𝐶(1 + |𝑥|𝐶) ,
𝑑𝑋 (𝑠) = 𝐴𝑋 (𝑠) 𝑑𝑠 + 𝐹 (𝑡, 𝑋𝑠 ) 𝑑𝑠 + 𝐺 (𝑠, 𝑋𝑠 ) 𝑑𝑊 (𝑠) ,
𝑋𝑡 = 𝑥 ∈ C.
We make the following assumptions.
Hypothesis 1. (i) The operator 𝐴 is the generator of a strongly
continuous semigroup {𝑒𝑡𝐴 , 𝑡 ≥ 0} of bounded linear
operators in the Hilbert space 𝐻. We denote by 𝑀 and 𝜔 two
constants such that |𝑒𝑡𝐴 | ≤ 𝑀𝑒𝜔𝑡 , for 𝑡 ≥ 0.
(ii) The mapping 𝐹: [0, ∞) × C → 𝐻 is measurable and
satisfies, for some constant 𝐿 > 0 and 0 ≤ 𝜃 < 1,
󵄨
󵄨󵄨 𝑠𝐴
󵄨󵄨𝑒 𝐹 (𝑡, 𝑥)󵄨󵄨󵄨 ≤ 𝐿𝑒𝜔𝑠 𝑠−𝜃 (1 + |𝑥|𝐶) ,
󵄨
󵄨
󵄨󵄨 𝑠𝐴
󵄨
󵄨󵄨𝑒 𝐹 (𝑡, 𝑥) − 𝑒𝑠𝐴 𝐹 (𝑡, 𝑦)󵄨󵄨󵄨 ≤ 𝐿𝑒𝜔𝑠 𝑠−𝜃 󵄨󵄨󵄨󵄨𝑥 − 𝑦󵄨󵄨󵄨󵄨𝐶,
󵄨
󵄨
(13)
𝑋𝑡 = 𝑥 ∈ C.
𝑠∈[𝑡,𝑇]
𝑠 ∈ [𝑡, ∞) , (10)
𝑠 ∈ [𝑡, ∞) ,
(14)
for some constant C depending only on 𝑞, 𝛾, 𝜃, T, 𝜏, L, 𝜔, and
M.
By Theorem 1 and the arbitrariness of 𝑇 in its statement,
the solution is defined for every 𝑠 ≥ 𝑡. We have the following
result.
Theorem 2. Assume that Hypothesis 1 holds and the process
𝑋(⋅, 𝑡, 𝑥) is mild solution of (10) with initial value (𝑡, 𝑥) ∈
[0, ∞) × C. Then, for every 𝑞 ∈ [1, ∞), there exists a constant
𝑞
𝜂(𝑞) such that the process 𝑋⋅ (𝑡, 𝑥) ∈ H𝜂(𝑞) (𝑡). Moreover, for a
suitable constant 𝐶 > 0, one has
∞
(11)
𝑠 > 0, 𝑡 ∈ [0, +∞) , 𝑥, 𝑦 ∈ C.
(iii) 𝐺 is a mapping [0, ∞) × C → 𝐿(Ξ, 𝐻) such that for
every 𝑣 ∈ Ξ, the map 𝐺𝑣: [0, ∞) × C → 𝐻 is measurable,
𝑞
󵄨 󵄨𝑞
󵄨 󵄨𝑞
𝐸sup 𝑒𝜂(𝑞)𝑞𝑠 󵄨󵄨󵄨𝑋𝑠 󵄨󵄨󵄨𝐶 + 𝐸 ∫ 𝑒𝜂(𝑞)𝑞𝑠 󵄨󵄨󵄨𝑋𝑠 󵄨󵄨󵄨𝐶𝑑𝑠 ≤ 𝐶(1 + |𝑥|𝐶) ,
s≥t
𝑡
(15)
with the constant 𝜂(𝑞) depending only on 𝑞, 𝛾, 𝜃, 𝜏, 𝐿, 𝜔, and
𝑀.
4
Abstract and Applied Analysis
Proof. We define a mapping Φ from H𝑞𝜂 (𝑡) × [0, ∞) × C to
H𝑞𝜂 (𝑡) by the formula
Φ(𝑋⋅ , 𝑡, 𝑥)𝑠 (𝑙) = 𝑒(𝑠+𝑙−𝑡)𝐴 𝑥 (0) + ∫
𝑠+𝑙
𝑡
+∫
𝑠+𝑙
𝑡
𝑠 ∈ [𝑡, ∞) ,
(𝑠+𝑙−𝜎)𝐴
𝑒
setting 𝑞󸀠 = 𝑞/(𝑞 − 1), so that
󵄨
󵄨𝑞
𝑒𝑞𝜂𝑠 󵄨󵄨󵄨󵄨Φ󸀠 (𝑋⋅ )𝑠 󵄨󵄨󵄨󵄨
𝑒(𝑠+𝑙−𝜎)𝐴 𝐹 (𝜎, 𝑋𝜎 ) 𝑑𝜎
𝑠+𝑙
≤ 𝑐𝛼𝑞 𝑀𝑞 sup 𝑒𝑞𝜂𝑠 (∫
𝐺 (𝜎, 𝑋𝜎 ) 𝑑𝑊 (𝜎) ,
𝑙 ∈ [−𝜏, 0] ,
𝑠+𝑙
𝑙 ∈ [−𝜏, 0] ,
−𝜏≤𝑙≤0
𝑠 + 𝑙 ≥ 𝑡,
𝑠 + 𝑙 < 𝑡.
For simplicity, we set 𝑡 = 0, and we treat only the case 𝐹 = 0,
the general case being handled in a similar way. We will use
the so called factorization method; see [28, Theorem 5.2.5].
Let us take 𝑞 > 1 and 𝛼 ∈ (0, 1) such that 1/𝑞 < 𝛼 < (1/2) −
s
𝛾, and let 𝑐𝛼−1 = ∫𝜎 (𝑠 − 𝑟)𝛼−1 (𝑟 − 𝜎)−𝛼 𝑑𝑟.
By the stochastic Fubini theorem,
+ 𝑐𝛼 ∫
0
𝑠+𝑙
∫
𝜎
−𝜏≤𝑙≤0
𝑙 ∈ [−𝜏, 0] ,
(18)
𝑙 ∈ [−𝜏, 0] ,
0
𝑟
0
(𝑠+𝑙)𝐴
󸀠
𝑠
× ∫ 𝑒(𝜂+𝜔)(𝑠−𝑟) 𝑒𝑞𝜂𝑟 |𝑌 (𝑟)|𝑞 𝑑𝑟.
0
(21)
Applying the Young inequality for convolutions, we have
∞
𝑞/𝑞
∞
󸀠
󵄨𝑞
󵄨󵄨Φ󸀠 (𝑋⋅ )𝑠 󵄨󵄨󵄨 𝑑𝑠 ≤ 𝑐𝛼𝑞 𝑀𝑞 (∫ 𝑒(𝜂+𝜔)𝑠 𝑠𝑞 (𝛼−1) 𝑑𝑠)
󵄨
󵄨
0
𝑞𝜂𝑠 󵄨󵄨
∫ 𝑒
∞
∞
0
0
󸀠
and we conclude that
󵄨󵄨
󵄨󵄨 󸀠
󵄨󵄨󵄨Φ (𝑋⋅ )󵄨󵄨󵄨𝐿𝑞 (Ω;𝐿𝑞𝜂 (C)) ≤ 𝑐𝛼 𝑀|𝑌|𝐿𝑞P (Ω;𝐿𝑞𝜂 (𝐻))
P
∞
𝑠 + 𝑙 < 0,
1/𝑞󸀠
󸀠
1/𝑞
0
(23)
.
If we start again from (20) and apply the Hölder inequality, we
obtain
𝐺 (𝜎, 𝑋𝜎 ) 𝑑𝑊 (𝜎) .
𝜔𝑠
(22)
× (∫ 𝑒(𝜂+𝜔)𝑠 𝑑𝑠)
(19)
𝑌 (𝑟) = ∫ (𝑟 − 𝜎) 𝑒
≤
𝑞/𝑞
𝑠
󸀠
𝑐𝛼𝑞 𝑀𝑞 (∫ 𝑒(𝜂+𝜔)𝑟 𝑟𝑞 (𝛼−1) 𝑑𝑟)
0
0
(𝑠 + 𝑙 − 𝑟)𝛼−1 𝑒(𝑠+𝑙−𝑟)𝐴 𝑌 (𝑟) 𝑑𝑟,
−𝛼 (𝑟−𝜎)𝐴
0
× (∫ 𝑒(𝜂+𝜔)𝑠 𝑠𝑞 (𝛼−1) 𝑑𝑠)
where
Φ󸀠 (𝑋⋅ )𝑠 (𝑙) = 𝑐𝛼 ∫
𝑠+𝑙
× ∫ 𝑒(𝜂+𝜔)(𝑠−𝑟) 𝑒𝑞𝜂𝑟 |𝑌 (𝑟)|𝑞 𝑑𝑟
∞
𝑠 + 𝑙 ≥ 0,
Φ(𝑋⋅ , 0, 𝑥)𝑠 (𝑙) = 𝑥 (𝑠 + 𝑙) ,
𝑠+𝑙
𝑞/𝑞󸀠
0
(𝑠 + 𝑙 − 𝑟)𝛼−1 (𝑟 − 𝜎)−𝛼
= 𝑒(𝑠+𝑙)𝐴 𝑥 (0) + Φ󸀠 (𝑋𝑠 ) (𝑙) ,
𝑠 ∈ [0, ∞) ,
󸀠
× ∫ 𝑒(𝜂+𝜔)𝑠 𝑑𝑠 ∫ 𝑒𝑞𝜂𝑠 |𝑌 (𝑠)|𝑞 𝑑𝑠,
× 𝑒(𝑠+𝑙−𝑟)𝐴 𝑒(𝑟−𝜎)𝐴 𝑑𝑟𝐺 (𝜎, 𝑋𝜎 ) 𝑑𝑊 (𝜎)
𝑠 ∈ [0, ∞) ,
𝑞
𝑠+𝑙
0
𝑥 (0)
𝑠+𝑙
0
≤ 𝑐𝛼𝑞 𝑀𝑞 sup (∫ 𝑒(𝜂+𝜔)(𝑠+𝑙−𝑟) (𝑠 + 𝑙 − 𝑟)(𝛼−1)𝑞 𝑑𝑟)
We are going to show that, provided 𝜂 is suitably chosen,
Φ(⋅, 𝑡, 𝑥) is well defined and that it is a contraction in H𝑞𝜂 (𝑡);
that is, there exists 𝑐 < 1 such that
󵄨󵄨
󵄨
󵄨󵄨Φ (𝑋⋅1 , 𝑡, 𝑥) − Φ (𝑋⋅1 , 𝑡, 𝑥)󵄨󵄨󵄨 𝑞
󵄨
󵄨H𝜂 (𝑡)
(17)
󵄨
󵄨
≤ 𝑐󵄨󵄨󵄨󵄨𝑋⋅1 − 𝑋⋅2 󵄨󵄨󵄨󵄨H𝑞 (𝑡) , 𝑋⋅1 , 𝑋⋅2 ∈ H𝑞𝜂 (𝑡) .
𝜂
Φ(𝑋⋅ , 0, 𝑥)𝑠 (𝑙) = 𝑒
󸀠
× 𝑒((𝜔+𝜂)/𝑞)(𝑠−𝑟) 𝑒𝜂𝑟 |𝑌 (𝑟)| 𝑑𝑟)
(16)
(𝑠+𝑙)𝐴
𝑞
≤ 𝑐𝛼𝑞 𝑀𝑞 sup (∫ (𝑠 + 𝑙 − 𝑟)𝛼−1 𝑒((𝜔+𝜂)/𝑞 )(𝑠+𝑙−𝑟)
Φ(𝑋⋅ , 𝑡, 𝑥)𝑠 (𝑙) = 𝑥 (𝑠 + 𝑙 − 𝑡) ,
𝑠 ∈ [𝑡, ∞) ,
0
−𝜏≤𝑙≤0
(𝑠+𝑙− 𝑟)𝛼−1 𝑒𝜔(𝑠+𝑙−𝑟) |𝑌 (𝑟)| 𝑑𝑟)
(𝑠+⋅)𝐴
Since sup−𝜏≤𝑙≤0 |𝑒
𝑥(0)| ≤ 𝑀𝑒 |𝑥|𝐶, the process 𝑒
𝑥(0), 𝑠 ≥ 0, belongs to H𝑞𝜂 provided 𝜔 + 𝜂 < 0. Next, we
estimate Φ󸀠 (𝑋⋅ ), where
𝑠+𝑙
󵄨󵄨 󸀠
󵄨
󵄨󵄨Φ (𝑋⋅ )𝑠 (𝑙)󵄨󵄨󵄨 ≤ 𝑐𝛼 ∫ (𝑠 + 𝑙 − 𝑟)𝛼−1 𝑀𝑒(𝑠+𝑙−𝑟)𝜔 |𝑌 (𝑟)| 𝑑𝑟,
󵄨
󵄨
0
(20)
𝑠+𝑙
󸀠
󸀠
󵄨󵄨
󵄨󵄨 𝜂(𝑠+𝑙) 󸀠
󵄨󵄨 ≤ 𝑐𝛼 𝑀(∫ 𝑟(𝛼−1)𝑞 𝑒(𝜔+𝜂)𝑟𝑞 𝑑𝑟)
󵄨󵄨𝑒
Φ
(𝑋
)
(𝑙)
⋅
𝑠
󵄨
󵄨
0
× (∫
𝑠+𝑙
0
𝑒𝜂𝑟𝑞 |𝑌 (𝑟)|𝑞 𝑑𝑟)
1/𝑞
,
(24)
𝑠
󸀠
󸀠
󵄨
󵄨󵄨 𝜂𝑠 󸀠
󵄨󵄨𝑒 Φ (𝑋⋅ )𝑠 󵄨󵄨󵄨 ≤ 𝑐𝛼 𝑀(∫ 𝑟(𝛼−1)𝑞 𝑒(𝜔+𝜂)𝑟𝑞 𝑑𝑟)
󵄨
󵄨
0
𝑠
× (∫ 𝑒𝜂𝑟𝑞 |𝑌(𝑟)|𝑞 𝑑𝑟)
0
1/𝑞󸀠
1/𝑞
.
1/𝑞󸀠
Abstract and Applied Analysis
5
So, we conclude that
󵄨󵄨 󸀠
󵄨
𝑞
𝑞
󵄨󵄨Φ (𝑋⋅ )󵄨󵄨󵄨 𝑞
󵄨
󵄨𝐿 P (Ω;𝐶𝜂 (C)) ≤ 𝑐𝛼 𝑀|𝑌|𝐿 P (Ω;𝐿 𝜂 (𝐻))
∞
󸀠
󸀠
× (∫ 𝑟(𝛼−1)𝑞 𝑒(𝜔+𝜂)𝑟𝑞 𝑑𝑟)
1/𝑞󸀠
0
(25)
.
On the other hand, by the Burkholder-Davis-Gundy inequalities, for some constant 𝑐𝑞 depending only on 𝑞, we have
𝑟
󵄨
󵄨2
𝐸|𝑌 (𝑟)|𝑞 ≤ 𝑐𝑞 𝐸(∫ (𝑟 − 𝜎)−2𝛼 󵄨󵄨󵄨󵄨𝑒(𝑟−𝜎)𝐴 𝐺 (𝜎, 𝑋𝜎 )󵄨󵄨󵄨󵄨𝐿
0
2 (Ξ,𝐻)
𝑑𝜎)
𝑞/2
𝑟
󵄨 󵄨2
× (∫ (𝑟 − 𝜎)−2𝛼−2𝛾 𝑒2𝜔(𝑟−𝜎) (1 + 󵄨󵄨󵄨𝑋𝜎 󵄨󵄨󵄨𝐶) 𝑑𝜎)
0
𝑞/2
,
(26)
which implies that
[𝐸|𝑌 (𝑟)| ]
≤
𝐿2 𝑐𝑞2/𝑞
𝑆 (𝑠) = 𝐼,
for 𝑠 < 0,
(31)
and we consider the equation
𝑋 (𝑠) = 𝑆 (𝑠 − 𝑡) 𝑥 ((0 ∧ (𝑠 − 𝑡)) ∨ (−𝜏))
∫ (𝑟 − 𝜎)
+ ∫ 𝐼[𝑡,∞) (𝜎) 𝑆 (𝑠 − 𝜎) 𝐹 (𝜎, 𝑋𝜎 ) 𝑑𝜎
−2𝛼−2𝛾
0
0
(27)
𝑠
(32)
+ ∫ 𝐼[𝑡,∞) (𝜎) 𝑆 (𝑠 − 𝜎)
0
so that
2/𝑞
for 𝑠 ≥ 0,
𝑠
𝑟
󵄨 󵄨 𝑞 2/𝑞
× 𝑒2𝜔(𝑟−𝜎) [𝐸(1 + 󵄨󵄨󵄨𝑋𝜎 󵄨󵄨󵄨𝐶) ] 𝑑𝜎,
𝑒2𝜂𝑟 [𝐸|𝑌 (𝑟)|𝑞 ]
For investigating the dependence of the solution 𝑋(𝑠, 𝑡, 𝑥)
on the initial data 𝑥 and 𝑡, we reformulate (13) as an equation
on [0, ∞). We set
𝑆 (𝑠) = 𝑒𝑠𝐴 ,
≤ 𝐿𝑞 𝑐𝑞 𝐸
𝑞 2/𝑞
Recalling the inequalities (23) and (25) and noting that the
map Y → Φ󸀠 (X⋅ ) is linear, we obtain an explicit expression
for the constant 𝑐 in (17), and it is immediate to verify that
𝑐 < 1 provided 𝜂 < 0 is chosen sufficiently large. We fix
such a value of 𝜂(𝑞). The first result is a consequence of the
contraction principle. The estimate (15) also follows from the
contraction property of Φ(⋅, 𝑡, 𝑥).
𝑟
≤ 𝐶1 ∫ (𝑟 − 𝜎)−2𝛼−2𝛾 𝑒2(𝜔+𝜂)(𝑟−𝜎) 𝑒2𝜂𝜎 𝑑𝜎
× 𝐺 (𝜎, 𝑋𝜎 ) 𝑑𝑊 (𝜎) ,
𝑠 ∈ [0, ∞) ,
0
𝑟
𝑋0 (𝜃) = 𝑥 ((−𝑡 + 𝜃) ∨ (−𝜏)) ,
+ 𝐶2 ∫ (𝑟 − 𝜎)−2𝛼−2𝛾 𝑒2(𝜔+𝜂)(𝑟−𝜎)
𝜃 ∈ [−𝜏, 0] .
0
󵄨 󵄨𝑞 2/𝑞
× 𝑒2𝜂𝜎 [𝐸󵄨󵄨󵄨𝑋𝜎 󵄨󵄨󵄨𝐶] 𝑑𝜎,
(28)
for suitable constants 𝐶1 , 𝐶2 . Applying the Young inequality
for convolutions, we obtain
∞
∞
0
0
𝑞/2
∫ 𝑒𝑞𝜂𝑟𝐸|𝑌 (𝑟)|𝑞 𝑑𝑠 ≤ 𝐶1 (∫ 𝑠−2𝛼−2𝛾 𝑒2(𝜔+𝜂)𝑠 𝑑𝑠)
∞
+ 𝐶2 (∫ 𝑠−2𝛼−2𝛾 𝑒2(𝜔+𝜂)𝑠 𝑑𝑠)
∞
∫ 𝑒𝑞𝜂𝑠 𝑑𝑠
0
𝑞/2
Lemma 3 (Parameter Depending Contraction Principle). Let
𝐵, 𝐷 denote Banach spaces. Let ℎ : 𝐵×𝐷 → 𝐵 be a continuous
mapping satisfying
0
∞
󵄨 󵄨𝑞
× ∫ 𝑒𝑞𝜂𝑠 𝐸󵄨󵄨󵄨𝑋𝑠 󵄨󵄨󵄨𝐶𝑑𝑠.
0
(29)
This shows that |𝑌|𝐿𝑞P (Ω;𝐿𝑞𝜂 (𝐻)) is finite provided we assume
that 𝜂 < 0 and 𝜔 + 𝜂 < 0; so, the map is well defined.
If 𝑋⋅1 , 𝑋⋅2 are processes belonging to H𝑞𝜂 and 𝑌1 , 𝑌2 are
defined accordingly, the entirely analogous passages show
that
1/𝑞 󵄨󵄨 1
2 󵄨󵄨
󵄨󵄨󵄨𝑌1 − 𝑌2 󵄨󵄨󵄨 𝑞
󵄨󵄨𝐿 (Ω;𝐿𝑞𝜂 (𝐻)) ≤ 𝐿𝑐𝛼 󵄨󵄨󵄨𝑋⋅ − 𝑋⋅ 󵄨󵄨󵄨𝐿𝑞 (Ω;𝐿𝑞𝜂 (C))
󵄨󵄨
P
P
(30)
1/2
∞
× (∫ 𝑠−2𝛼−2𝛾 𝑒2(𝜔+𝜂)𝑠 𝑑𝑠) .
0
Under the assumptions of Hypothesis 1, by Theorem 2, it is
easy to prove that equation (32) has a unique solution 𝑋 and
𝑞
𝑋⋅ ∈ H𝜂(𝑞) for every 𝑞 ∈ [2, ∞). It clearly satisfies 𝑋(𝑠) =
𝑥((𝑠 − 𝑡) ∨ (−𝜏)) for 𝑠 ∈ [−𝜏, 𝑡), and its restriction to the time
internal [𝑡, ∞) is the unique mild solution of (10). From now
on, we denote by 𝑋(𝑠, 𝑡, 𝑥), 𝑠 ∈ [0, ∞), the solution of (32).
We need the following parameter-depending contraction
principle, which is stated in the following lemma and proved
in [29, Theorems 10.1 and 10.2].
󵄨
󵄨
󵄨
󵄨󵄨
󵄨󵄨ℎ (𝑥1 , 𝑦) − ℎ (𝑥2 , 𝑦)󵄨󵄨󵄨 ≤ 𝛼 󵄨󵄨󵄨𝑥1 − 𝑥2 󵄨󵄨󵄨 ,
(33)
for some 𝛼 ∈ [0, 1) and every 𝑥1 , 𝑥2 ∈ 𝐵, y ∈ 𝐷. Let 𝜙(𝑦)
denote the unique fixed point of the mapping ℎ(⋅, 𝑦) : 𝐵 → 𝐵.
Then, 𝜙 : 𝐷 → 𝐵 is continuous.
Theorem 4. Assume that Hypothesis 1 holds true. Then, for
every 𝑞 ∈ [1, ∞), the map (𝑡, 𝑥) → 𝑋⋅ (𝑡, 𝑥) is continuous
q
from [0, ∞) × C to H𝜂(q) .
Proof. Clearly, it is enough to prove the claim for 𝑞 large. Let
us consider the map Φ defined in the proof of Theorem 2. In
6
Abstract and Applied Analysis
our present notation, Φ can be seen as a mapping from H𝑞𝜂 ×
[0, ∞) × C to H𝑞𝜂 as follows:
Φ(𝑋⋅ , 𝑡, 𝑥)𝑠 (𝑙) = 𝑆 (𝑠 + 𝑙 − 𝑡) 𝑥 (0)
𝑠+𝑙
+∫
0
𝑠+𝑙
+∫
0
𝐼[𝑡,∞) (𝜎) 𝑆 (𝑠 + 𝑙 − 𝜎) 𝐹 (𝜎, 𝑋𝜎 ) 𝑑𝜎
𝐼[𝑡,∞) (𝜎) 𝑆 (𝑠 + 𝑙 − 𝜎)
for 𝑠 varying on the time interval [𝑡, ∞) ⊂ [0, ∞). As
in Section 2, we extend the domain of the solution setting
𝑋(𝑠, 𝑡, 𝑥) = 𝑥((𝑠 − 𝑡) ∨ (−𝜏)) for 𝑠 ∈ [−𝜏, 𝑡).
We make the following assumptions.
Hypothesis 2. The mapping 𝜓 : [0, ∞) × C × 𝐾 × 𝐿 2 (Ξ, 𝐾) →
𝐾 is Borel measurable such that, for all 𝑡 ∈ [0, ∞), 𝜓(𝑡, ⋅) : C×
𝐾 × 𝐿 2 (Ξ, 𝐾) → 𝐾 is continuous, and for some 𝐿 𝑦 , 𝐿 𝑧 > 0,
𝜇 ∈ 𝑅, and 𝑚 ≥ 1,
󵄨󵄨
󵄨
󵄨󵄨𝜓 (𝑠, 𝑥, 𝑦1 , 𝑧1 ) − 𝜓 (𝑠, 𝑥, 𝑦2 , 𝑧2 )󵄨󵄨󵄨
󵄨
󵄨
󵄨
󵄨
≤ 𝐿 𝑦 󵄨󵄨󵄨𝑦1 − 𝑦2 󵄨󵄨󵄨 + 𝐿 𝑧 󵄨󵄨󵄨𝑧1 − 𝑧2 󵄨󵄨󵄨 ,
󵄨
󵄨󵄨
󵄨 󵄨
𝑚
󵄨󵄨𝜓 (𝑠, 𝑥, 𝑦, 𝑧)󵄨󵄨󵄨 ≤ 𝐿 (1 + |𝑥|𝐶 + 󵄨󵄨󵄨𝑦󵄨󵄨󵄨 + |𝑧|) ,
× 𝐺 (𝜎, 𝑋𝜎 ) 𝑑𝑊 (𝜎) ,
𝑠 ∈ [0, ∞) ,
𝑙 ∈ [−𝜏, 0] ,
𝑠 + 𝑙 ≥ 𝑡,
Φ(𝑋⋅ , 𝑡, 𝑥)𝑠 (𝑙) = 𝑥 ((𝑠 + 𝑙 − 𝑡) ∨ (−𝜏)) ,
𝑠 ∈ [0, ∞) ,
𝑙 ∈ [−𝜏, 0] ,
𝑠 + 𝑙 ≤ 𝑡.
(34)
By the arguments of the proof of Theorem 2, Φ(⋅, 𝑡, 𝑥)
is a contraction in H𝑞𝜂 uniformly with respect to 𝑡, 𝑥.
The process 𝑋⋅ (𝑡, 𝑥) is the unique fixed point of Φ(⋅, 𝑡, 𝑥).
So, by the parameter-depending contraction principle
(Lemma 3), it suffices to show that Φ is continuous from
H𝑞𝜂 × [0, ∞) × C to H𝑞𝜂 . From the contraction property
of Φ(⋅, 𝑡, 𝑥) mentioned earlier, we have that Φ(⋅, 𝑡, 𝑥) is
continuous, uniformly in 𝑡, 𝑥. Moreover, for fixed 𝑋⋅ , it is
easy to verify that Φ(𝑋⋅ , ⋅, ⋅) is continuous from [0, ∞) × C
to H𝑞𝜂 . The proof is finished.
Remark 5. By similar passages, we can show that, for fixed
𝑡, Theorem 4 still holds true for 𝑞 large enough if the spaces
[0, ∞) × C and H𝑞𝜂 are replaced by the spaces 𝐿𝑞 (Ω, C, F𝑡 )
and H𝑞𝜂 (𝑡) respectively, where 𝐿𝑞 (Ω, C, F𝑡 ) denotes that the
space of F𝑡 -measurable function with value in C, such that
the norm
𝑞
|𝑥|𝑞 = 𝐸|𝑥|𝐶,
(35)
󵄨2
󵄨
⟨𝜓 (𝑠, 𝑥, 𝑦1 , 𝑧) − 𝜓 (𝑠, 𝑥, 𝑦2 , 𝑧) , 𝑦1 − 𝑦2 ⟩𝐾 ≥ 𝜇󵄨󵄨󵄨𝑦1 − 𝑦2 󵄨󵄨󵄨 ,
(37)
for every 𝑠 ∈ [0, ∞), 𝑥 ∈ C, 𝑦, 𝑦1 , 𝑦2 ∈ 𝐾, 𝑧, 𝑧1 , and 𝑧2 ∈
𝐿 2 (Ξ, 𝐾).
We note that the third inequality in (37) follows from the
first one taking 𝜇 = −𝐿 𝑦 but that the third inequality may also
hold for different values of 𝜇.
Firstly, we consider the backward stochastic differential
equation
𝑇
𝑇
𝑠
𝑠
𝑌 (𝑠) − 𝑌 (𝑇) + ∫ 𝑍 (𝜎) 𝑑𝑊 (𝜎) + 𝜆 ∫ 𝑌 (𝜎) 𝑑𝜎
𝑇
= ∫ 𝜓 (𝜎, 𝑋𝜎 , 𝑌 (𝜎) , 𝑍 (𝜎)) 𝑑𝜎,
𝑠
0 ≤ 𝑠 ≤ 𝑇 < ∞.
(38)
𝐾 is a Hilbert space, the mapping 𝜓 : [0, ∞) × C × 𝐾 ×
𝐿 2 (Ξ, 𝐾) → 𝐾 is a given measurable function, 𝑋⋅ is a
predictable process with values in another Banach space C,
and 𝜆 is a real number.
Theorem 6. Assume that Hypothesis 2 holds. Let 𝑝 > 2 and
𝛿 < 0 be given, and choose
is finite.
3. The Backward-Forward System
𝑞 ≥ 𝑚𝑝,
In this section, we consider the system of stochastic differential equations, 𝑃-a.s.,
𝑠
𝑡
+ ∫ 𝑒(𝑠−𝜎)𝐴 𝐺 (𝜎, 𝑋𝜎 ) 𝑑𝑊 (𝜎) ,
𝑡
𝑠 ∈ [𝑡, ∞) ,
𝑋𝑡 = 𝑥 ∈ C,
𝑇
𝑠
𝑠
= ∫ 𝜓 (𝜎, 𝑋𝜎 , 𝑌 (𝜎) , 𝑍 (𝜎)) 𝑑𝜎,
𝑠
(i) For 𝑋⋅ ∈ 𝐿 P (Ω; 𝐿𝑞𝜂 (C)) and 𝜆 > −(𝛿 + 𝜇 − (𝐿2𝑧 /2)),
p
(38) has a unique solution in K𝛿 that will be denoted
by (𝑌(𝑋⋅ )(𝑠), 𝑍(𝑋⋅ )(𝑠)), 𝑠 ≥ 0.
(ii) The estimate
∞
𝑇
(39)
Then, the following hold.
𝑝
󵄨
󵄨2
𝐸sup(𝑌 (𝑋⋅ ) (𝑠)) 𝑒𝑝𝛿𝑠 + 𝐸(∫ 𝑒2𝛿𝜎 󵄨󵄨󵄨𝑌(𝑋⋅ )(𝜎)󵄨󵄨󵄨 𝑑𝜎)
∞
0 ≤ 𝑠 ≤ 𝑇 < ∞,
(36)
𝑝/2
0
𝑠≥0
𝑌 (𝑠) − 𝑌 (𝑇) + ∫ 𝑍 (𝜎) 𝑑𝑊 (𝜎) + 𝜆 ∫ 𝑌 (𝜎) 𝑑𝜎
𝑇
𝛿
.
𝑚
𝑞
𝑋 (𝑠) = 𝑒(𝑠−𝑡)𝐴 𝑥 (0) + ∫ 𝑒(𝑠−𝜎)𝐴 𝐹 (𝜎, 𝑋𝜎 ) 𝑑𝜎
𝑠
𝜂>
󵄨
󵄨2
+ 𝐸(∫ 𝑒2𝛿𝜎 󵄨󵄨󵄨𝑍 (𝑋⋅ ) (𝜎)󵄨󵄨󵄨 𝑑𝜎)
0
𝑝
󵄨 󵄨𝑚
≤ 𝑐(1 + 󵄨󵄨󵄨𝑋⋅ 󵄨󵄨󵄨𝐿𝑞 (Ω;𝐿𝑞𝜂 (C)) )
P
𝑝/2
(40)
Abstract and Applied Analysis
7
holds for a suitable constant 𝑐. In particular, 𝑌(𝑋⋅ ) ∈
𝑝
𝐿 P (Ω; 𝐶𝛿 (𝐾)).
(iii) The map 𝑋⋅ → (𝑌(𝑋⋅ ), 𝑍(𝑋⋅ )) is continuous from
𝑞
𝑝
𝐿 P (Ω; 𝐿𝑞𝜂 (C)) to K𝛿 , and 𝑋⋅ → 𝑌(𝑋⋅ ) is continuous
𝑞
𝑝
from 𝐿 P (Ω; 𝐿𝑞𝜂 (C)) to 𝐿 P (Ω; 𝐶𝛿 (𝐾)).
(iv) The statements of points (i), (ii), and (iii) still hold
𝑞
true if the space 𝐿 P (Ω; 𝐿𝑞𝜂 (C)) is replaced by the space
𝑞
𝐿 P (Ω; 𝐶𝜂 (C)).
Proof. The theorem is very similar to Proposition 3.11 in [24].
The only minor difference is that the mapping 𝜓 : [0, ∞) ×
C × 𝐾 × 𝐿 2 (Ξ, 𝐾) → 𝐾 is a given measurable function, while
in [24], the measurable function 𝜓 is from 𝐻 × 𝐾 × 𝐿 2 (Ξ, 𝐾)
to 𝐾; however, the same arguments apply.
Theorem 7. Assume that Hypothesis 1 holds and that
Hypothesis 2 holds true in the particular case 𝐾 = 𝑅. Then,
for every 𝑝 > 2, 𝑞, 𝛿 < 0 satisfying (39) with 𝜂 = 𝜂(𝑞),
and for every 𝜆 > 𝜆󸀠 = −(𝛿 + 𝜇 − (𝐿2𝑧 /2)), there exists a
𝑞
𝑝
unique solution in H𝜂(𝑞) × K𝛿 of (36) that will be denoted
by (𝑋(⋅, 𝑡, 𝑥), 𝑌(⋅, 𝑡, 𝑥), 𝑍(⋅, 𝑡, 𝑥)). Moreover, 𝑌(⋅, 𝑡, 𝑥)
∈
𝑝
𝐿 P (Ω; 𝐶𝛿 (𝑅)). The map (𝑡, 𝑥) → (𝑌(⋅, 𝑡, 𝑥), 𝑍(⋅, 𝑡, 𝑥)) is con𝑝
tinuous from [0, ∞)×C to K𝛿 , and the map (𝑡, 𝑥) → 𝑌(⋅, 𝑡, 𝑥)
𝑝
is continuous from [0, ∞) × C to 𝐿 P (Ω; 𝐶𝛿 (𝑅)).
Proof. We first notice that the system is decoupled; the first
does not contain the solution (𝑌, 𝑍) of the second one. Therefore, under the assumption of Hypothesis 1 by Theorem 2,
𝑞
there exists a unique solution 𝑋(⋅, 𝑡, 𝑥) and 𝑋⋅ (𝑡, 𝑥) ∈ H𝜂(𝑞)
of the first equation. Moreover, from Theorem 4, it follows
that the map (𝑡, 𝑥) → 𝑋⋅ (𝑡, 𝑥) is continuous from [0, ∞) × C
𝑞
to H𝜂(𝑞) .
Let 𝐾 = 𝑅; from Theorem 6, we have that there
𝑝
exists a unique solution (𝑌(⋅, 𝑡, 𝑥), 𝑍(⋅, 𝑡, 𝑥)) ∈ K𝛿 of the
second equation, and the map 𝑋⋅ → (𝑌(𝑋⋅ ), 𝑍(𝑋⋅ )) is
𝑞
𝑝
continuous from H𝜂(𝑞) to K𝛿 while X⋅ → (Y(X⋅ )) is
𝑞
determined on an interval [𝑠, ∞) by the values of the process
𝑋⋅ on the same interval, for 𝑡 ≤ 𝑠 < ∞, we have, 𝑃-a.s.,
𝑌 (𝑙, 𝑠, 𝑋𝑠 (𝑡, 𝑥)) = 𝑌 (𝑙, 𝑡, 𝑥) ,
𝑍 (𝑙, 𝑠, 𝑋𝑠 (𝑡, 𝑥)) = 𝑍 (𝑙, 𝑡, 𝑥) ,
󵄨󵄨
󵄨
󵄨󵄨𝑓𝑛 (𝜔) − 𝑓 (𝜔)󵄨󵄨󵄨 ↓ 0 for all 𝜔 ∈ Ω as 𝑛 → ∞.
𝑋𝑙 (𝑠, 𝑋𝑠 (𝑡, 𝑥)) = 𝑋𝑙 (𝑡, 𝑥) ,
𝑙 ∈ [𝑠, ∞)
(41)
is a consequence of the uniqueness of the solution of (13).
Since the solution of the backward equation is uniquely
(43)
Hence, 𝑓𝑛 → 𝑓 in | ⋅ |𝐿𝑞 (Ω,C) by Lebesgue’s dominated
convergence theorem.
We are now in a position of showing the main result in
this section.
Theorem 10. Assume that Hypothesis 1 holds true and that
Hypothesis 2 holds in the particular case 𝐾 = 𝑅. Then, there
exist two Borel measurable deterministic functions 𝜐 : [t, ∞) ×
C → 𝑅 and 𝜁 : [𝑡, ∞) × C → Ξ∗ = 𝐿(Ξ, 𝑅) =
𝐿 2 (Ξ, 𝑅), such that for 𝑡 ∈ [0, ∞) and x ∈ C, the solution
(𝑋(𝑡, 𝑥), 𝑌(𝑡, 𝑥), 𝑍(𝑡, 𝑥)) of (36) satisfies
𝑌 (𝑠, 𝑡, 𝑥) = 𝜐 (𝑠, 𝑋𝑠 (𝑡, 𝑥)) ,
𝑍 (𝑠, 𝑡, 𝑥) = 𝜁 (𝑠, 𝑋𝑠 (𝑡, 𝑥)) ,
𝑃-a.s., for a.a. 𝑠 ∈ [𝑡, ∞) .
(44)
𝑝
Remark 8. From Remark 5, by similar passages, we can show
that for fixed 𝑡 and for 𝑞 large enough, under the assumptions
of Theorem 7, the map 𝑥 → (𝑌(⋅, 𝑡, 𝑥), 𝑍(⋅, 𝑡, 𝑥)) is continu𝑝
ous from 𝐿𝑞 (Ω, C, F𝑡 ) to K𝛿 (𝑡).
We also remark that the process 𝑋(⋅, 𝑡, 𝑥) is F[𝑡,∞)
measurable, since C is separable Banach space, we have that
𝑋⋅ (𝑡, 𝑥) is F[𝑡,∞) measurable; So that 𝑌(𝑡) is measurable
with respect to both F[𝑡,∞) and F𝑡 , it follows that 𝑌(𝑡) is
deterministic.
For later use, we notice three useful identities; for 𝑡 ≤ 𝑠 <
∞, the equality, 𝑃-a.s.,
(42)
Let now 𝑓 ∈ 𝐿𝑞 (Ω, C). By Lemma 9 we get the existence
of a sequence of simple function 𝑓𝑛 , 𝑛 ∈ 𝑁, such that
𝑝
𝑞
for a.a. 𝑙 ∈ [𝑠, ∞) .
Lemma 9 (see [30]). Let 𝐸 be a metric space with metric 𝑑,
and let 𝑓 : Ω → 𝐸 be strongly measurable. Then, there
exists a sequence 𝑓𝑛 , 𝑛 ∈ 𝑁, of simple 𝐸-valued functions
(i.e., 𝑓𝑛 is F/B(E) measurable and takes only a finite number
of values) such that for arbitrary 𝜔 ∈ Ω, the sequence
𝑑(𝑓𝑛 (𝜔), 𝑓(𝜔)), 𝑛 ∈ 𝑁, is monotonically decreasing to zero.
continuous from H𝜂(𝑞) to 𝐿 P (Ω; 𝐶𝛿 (𝑅)). We have proved that
(𝑋(⋅, 𝑡, 𝑥), 𝑌(⋅, 𝑡, 𝑥), 𝑍(⋅, 𝑡, 𝑥)) ∈ H𝜂(𝑞) × K𝛿 is the unique
solution of (36), and the other assertions follow from composition.
for 𝑙 ∈ [𝑠, ∞) ,
Proof. We apply the techniques introduced in [26, Proposition 3.2]. Let {𝑒𝑖 } be a basis of Ξ∗ , and let us define 𝑍𝑖,𝑁 =
((𝑍, 𝑒𝑖 )Ξ∗ ∧ 𝑁) ∨ (−𝑁). Then, for every 0 ≤ 𝑡1 < 𝑡2 < ∞, Δ >
0, and 𝑥1 , 𝑥2 ∈ C, we have that
󵄨󵄨
󵄨󵄨 𝑡1 +Δ
𝑡2 +Δ
󵄨󵄨
󵄨󵄨
𝑖,𝑁
𝑖,𝑁
󵄨󵄨
󵄨󵄨𝐸 ∫
𝑍
(𝑠,
𝑡
,
𝑥
)
𝑑𝑠
−
𝐸
𝑍
(𝑠,
𝑡
,
𝑥
)
𝑑𝑠
∫
1 1
2 2
󵄨󵄨
󵄨󵄨 𝑡
𝑡2
󵄨
󵄨 1
𝑡2
󵄨
󵄨
≤ 𝐸 ∫ 󵄨󵄨󵄨󵄨𝑍𝑖,𝑁 (𝑠, 𝑡1 , 𝑥1 )󵄨󵄨󵄨󵄨 𝑑𝑠
𝑡
1
𝑡1 +Δ
+ 𝐸∫
𝑡2
𝑡2 +Δ
+ 𝐸∫
𝑡1 +Δ
󵄨󵄨 𝑖,𝑁
󵄨
󵄨󵄨𝑍 (𝑠, 𝑡1 , 𝑥1 ) − 𝑍𝑖,𝑁 (𝑠, 𝑡2 , 𝑥2 )󵄨󵄨󵄨 𝑑𝑠
󵄨
󵄨
󵄨󵄨 𝑖,𝑁
󵄨
󵄨󵄨𝑍 (𝑠, 𝑡2 , 𝑥2 )󵄨󵄨󵄨 𝑑𝑠
󵄨
󵄨
8
Abstract and Applied Analysis
󵄨
󵄨
≤ 2𝑁 󵄨󵄨󵄨𝑡2 − 𝑡1 󵄨󵄨󵄨 + Δ1/2 𝑒−𝛿(𝑡1 +Δ)
∞
2𝛿𝑠 󵄨󵄨 𝑖,𝑁
󵄨󵄨𝑍
󵄨
×(𝐸(∫ 𝑒
0
𝑁𝑚
𝑖,𝑁
(𝑠, 𝑡1 , 𝑥1 )−𝑍
1/𝑝
󵄨2 𝑝/2
(𝑠, 𝑡2 , 𝑥2 )󵄨󵄨󵄨󵄨 𝑑𝑠) )
× (𝐸(∫ 𝑒
󵄨2
󵄨󵄨𝑍 (𝑠, 𝑡1 , 𝑥1 )−𝑍 (𝑠, 𝑡2 , 𝑥2 )󵄨󵄨󵄨 𝑑𝑠)
𝑝/2
1/𝑝
)
𝑚→∞
.
𝑡+Δ
𝑡+(1/𝑛)
inf 𝑛𝐸 ∫
(𝑡, 𝑥) = lim
𝑛→∞
𝑡
𝑠
𝑘
𝑘=1
= lim 𝐸𝐼𝐵 (𝐸 ∫
From Theorem 7, we have that the map (𝑡, 𝑥) → ∫𝑡
𝑍𝑖,𝑁(𝑠, 𝑡, 𝑥)𝑑𝑠 is continuous from [0, ∞) × C to 𝑅. By
Remark 8, we also have that, for fixed 𝑡, the map 𝑥 →
𝑡+Δ
𝐸 ∫𝑡 𝐸𝑍𝑖,𝑁(𝑠, 𝑡, 𝑥)𝑑𝑠 is continuous from 𝐿𝑞 (Ω, C, F𝑡 ) to 𝑅
for 𝑞 large enough. Let us define
𝜁
𝑘=1
𝑁𝑚
(45)
𝑖,𝑁
𝑘
𝑚→∞
2𝛿𝑠 󵄨󵄨
0
𝑚→∞
𝑠+(1/𝑛)
= lim 𝐸 (𝐼𝐵 ∑ 𝐼{𝑓𝑚 =ℎ(𝑚) } ) 𝐸 ∫
󵄨
󵄨
≤ 2𝑁 󵄨󵄨󵄨𝑡2 − 𝑡1 󵄨󵄨󵄨 + Δ1/2 𝑒−𝛿(𝑡1 +Δ)
∞
= lim ∑ 𝐸 (𝐼𝐵 𝐼{𝑓𝑚 =ℎ(𝑚) } ∫
𝑠+(1/𝑛)
𝑠
𝑠+(1/𝑛)
= lim ∫ (𝐸 ∫
𝑚→∞ 𝐵
𝑠
𝑠+(1/𝑛)
= ∫ (𝐸 ∫
𝑠
𝐵
𝑍𝑖,𝑁 (𝑙, 𝑠, ℎ𝑘(𝑚) ) 𝑑𝑙)
𝑠+(1/𝑛)
𝑠
𝑍𝑖,𝑁 (𝑙, 𝑠, 𝑦) 𝑑𝑙|𝑦=𝑓𝑚 )
𝑍𝑖,𝑁 (𝑙, 𝑠, 𝑦) 𝑑𝑙|𝑦=𝑓𝑚 ) 𝑑𝑃
𝑍𝑖,𝑁 (𝑙, 𝑠, 𝑦) 𝑑𝑙|𝑦=𝑋𝑠 ) 𝑑𝑃.
(49)
and we get that
𝜁𝑖,𝑁 (𝑠, 𝑋𝑠 (𝑡, 𝑥)) = lim inf 𝑛
𝑛→∞
𝑖,𝑁
𝑍
(𝑠, 𝑡, 𝑥) 𝑑𝑠,
× [𝐸 ∫
(46)
𝑠+(1/𝑛)
𝑠
𝑡 ∈ [0, ∞) , 𝑥 ∈ C.
𝑍𝑖,𝑁 (𝑙, 𝑠, 𝑦) 𝑑𝑙|𝑦=𝑋𝑠 (𝑡,𝑥) ]
𝑠+(1/𝑛)
= lim inf 𝑛𝐸 [ ∫
It is clear that 𝜁𝑖,𝑁 : [0, ∞) × C → 𝑅 is a Borel function.
We fix 𝑥 and 0 ≤ 𝑡 ≤ 𝑠 < ∞. For 𝑙 ∈ [𝑠, ∞), we
denote 𝐸[𝑍𝑖,𝑁(𝑙, 𝑠, 𝑦)]|𝑦=𝑋𝑠 (𝑡,𝑥) , the random variable obtained
by composing 𝑋𝑠 (𝑡, 𝑥) with the map 𝑦 → 𝐸[𝑍𝑖,𝑁(𝑙, 𝑠, 𝑦)].
By Lemma 9, there exists a sequence of C-valued F𝑠 measurable simple functions
𝑛→∞
Fix 𝑡 and 𝑥. Recalling that |𝑍𝑖,𝑁| ≤ 𝑁, by the Lebesgue
theorem on differentiation, it follows that 𝑃-a.s.
lim 𝑛 ∫
𝑁𝑚
𝑓𝑚 : Ω 󳨀→ C, 𝑓𝑚 = ∑ ℎ𝑘(𝑚) 𝐼{𝑓𝑚 =ℎ(𝑚) } ,
𝑘
𝑘=1
𝑛→∞
𝑁𝑚 ∈ 𝑁,
(𝑚)
are pairwise distinct and Ω =
where ℎ1(𝑚) , . . . , ℎ𝑚
(𝑚)
ℎ𝑘 }, such that
for all 𝜔 ∈ Ω as 𝑛 󳨀→ ∞.
𝑠+(1/𝑛)
𝐵 𝑠
=
for a.a. 𝑠 ∈ [𝑡, ∞) .
By the boundedness of 𝑍𝑖,𝑁, applying the dominated convergence theorem, we get that
󵄨
𝜁𝑖,𝑁 (𝑠, 𝑋𝑠 (𝑡, 𝑥)) = 𝐸 [ 𝑍𝑖,𝑁 (𝑠, 𝑡, 𝑥)󵄨󵄨󵄨󵄨 F𝑠 ] = 𝑍𝑖,𝑁 (𝑠, 𝑡, 𝑥) ,
𝑃-a.s., for a.a. 𝑠 ∈ [𝑡, ∞) .
(52)
(48)
𝜁𝑖,𝑁 (𝑠, 𝑋𝑠 (𝑡, 𝑥)) = 𝑍𝑖,𝑁 (𝑠, 𝑡, 𝑥) ,
𝑍𝑖,𝑁 (𝑙, 𝑡, 𝑥) 𝑑𝑙𝑑𝑃
=∫ ∫
𝑠+(1/𝑛)
𝐵 𝑠
= 𝐸𝐼𝐵 ∫
𝑍
𝑠+(1/𝑛)
𝑠
𝑖,𝑁
𝑍𝑖,𝑁 (𝑙, 𝑠, 𝑋𝑠 ) 𝑑𝑙
= lim 𝐸 (𝐼𝐵 ∫
𝑚→∞
𝑃-a.s. for a.a. 𝑠 ∈ [𝑡, ∞) ,
(𝑙, 𝑠, 𝑋𝑠 ) 𝑑𝑙𝑑𝑃
𝑠+(1/𝑛)
𝑠
(51)
Now, we have proved that for every 𝑡, 𝑥,
For any 𝐵 ∈ F𝑠 , we have
∫ ∫
𝑠
𝑍𝑖,𝑁 (𝑙, 𝑡, 𝑥) 𝑑𝑙 = 𝑍𝑖,𝑁 (𝑠, 𝑡, 𝑥) ,
(47)
𝑁𝑚
{𝑓𝑚
⋃𝑘=1
𝑠
󵄨󵄨
󵄨
𝑍𝑖,𝑁 (𝑙, 𝑡, 𝑥) 𝑑𝑙󵄨󵄨󵄨󵄨 F𝑠 ] ,
󵄨󵄨
𝑃-a.s.
(50)
𝑠+(1/𝑛)
󵄨
󵄨󵄨
󵄨󵄨𝑓𝑚 (𝜔) − 𝑋𝑠 (𝜔)󵄨󵄨󵄨 ↓ 0
𝑍𝑖,𝑁 (𝑙, 𝑠, ℎ𝑘(𝑚) ) 𝑑𝑙
for every 𝑖, 𝑁. Let 𝐶 ⊂ [0, ∞) × C denote the set of
pairs (𝑡, 𝑥) such that lim𝑁 → ∞ 𝜁𝑖,𝑁(𝑡, 𝑥) exists and the series
𝑖,𝑁
∗
∑∞
𝑖=1 (lim𝑁 → ∞ 𝜁 (𝑡, 𝑥))𝑒𝑖 converges in Ξ . We define
∞
𝜁 (𝑡, 𝑥) = ∑ ( lim 𝜁𝑖,𝑁 (𝑡, 𝑥)) 𝑒𝑖 ,
𝑖=1
𝑖,𝑁
𝑍
(𝑙, 𝑠, 𝑓𝑚 ) 𝑑𝑙)
(53)
𝜁 (𝑡, 𝑥) = 0,
𝑁→∞
(𝑡, 𝑥) ∉ 𝐶.
(𝑡, 𝑥) ∈ 𝐶,
(54)
Abstract and Applied Analysis
9
Since 𝑍 satisfies
∞
𝑍 (𝜔, 𝑠, 𝑡, 𝑥) = ∑ ( lim 𝑍𝑖,𝑁 (𝜔, 𝑠, 𝑡, 𝑥)) 𝑒𝑖 ,
𝑖=1
𝑁→∞
(55)
for every 𝜔, 𝑠, 𝑡, 𝑥. From (53), it follows that for every 𝑡, 𝑥, we
have (𝑠, 𝑋𝑠 (𝜔, 𝑡, 𝑥)) ∈ 𝐶, 𝑃-a.s., for almost all 𝑠 ∈ [𝑡, ∞), and
𝑍(𝑠, 𝑡, 𝑥) = 𝜁(𝑠, 𝑋𝑠 (𝑡, 𝑥)) 𝑃-a.s., for a.a. 𝑠 ∈ [𝑡, ∞).
We define 𝜐(𝑡, 𝑥) = 𝑌(𝑡, 𝑡, 𝑥); since 𝑌(𝑡, 𝑡, 𝑥) is deterministic, so the map (𝑡, 𝑥) → 𝜐(𝑡, 𝑥) can be written as a
composition 𝜐(𝑡, 𝑥) = Γ3 (Γ2 (𝑡, Γ1 (𝑡, 𝑥))) with
Γ1 : [0, ∞) × C 󳨀→
𝑝
𝐿P
(Ω; 𝐶𝛿 (𝑅)) ,
𝑝
Γ2 : [0, ∞) × 𝐿 P (Ω; 𝐶𝛿 (𝑅)) 󳨀→ 𝐿𝑝 (Ω, 𝑅) ,
(56)
Γ2 (𝑡, 𝑉) = 𝑉 (𝑡) ,
Γ3 𝜉 = 𝐸𝜉.
From Theorem 7, it follows that Γ1 is continuous. By
|𝑉(𝑡) − 𝑈(𝑠)|𝐿𝑝 (Ω,𝑅) ≤ |𝑉 (𝑡) − 𝑉 (𝑠)|𝐿𝑝 (Ω,𝑅)
+ 𝑒−𝛿𝑝𝑠 |𝑉 − 𝑈|| 𝑝
𝐿
P
(Ω;𝐶𝛿 (𝑅))
,
(57)
we have that Γ2 is continuous. It is clear that Γ3 is continuous.
Then, the map (𝑡, 𝑥) → 𝜐(𝑡, 𝑥) is continuous from [0, ∞)×C
to 𝑅; therefore, 𝜐(𝑡, 𝑥)is a Borel measurable function. From
uniqueness of the solution of (36), it follows that 𝑌(𝑠, 𝑡, 𝑥) =
𝜐(𝑠, 𝑋𝑠 (𝑡, 𝑥)), 𝑃-a.s., for a.a. 𝑠 ∈ [𝑡, ∞).
Let (Ω, F, 𝑃) be a given complete probability space with a
filtration {F𝑡 }𝑡≥0 satisfying the usual conditions. {𝑊(𝑡), 𝑡 ≥ 0}
is a cylindrical Wiener process in Ξ with respect to {F𝑡 }𝑡≥0 .
We will say that an {F}𝑡≥0 -predictable process 𝑢 with values
in a given measurable space (𝑈, U) is an admissible control.
The function 𝑅 : [0, ∞) × C × 𝑈 → Ξ is measurable
and bounded. We consider the following controlled state
equation:
𝑢
𝐹 (𝑠, 𝑋𝑠𝑢 ) 𝑑𝑠
𝑑𝑋 (𝑠) = 𝐴𝑋 (𝑠) 𝑑𝑠 +
𝑠 ∈ [𝑡, ∞) ,
(58)
Here, we assume that there exists a mild solution of (58)
which will be denoted by 𝑋𝑢 (𝑠, 𝑡, 𝑥) or simply by 𝑋𝑢 (𝑠). We
consider a cost function of the form:
∞
𝐽 (𝑢) = 𝐸 ∫ 𝑒−𝜆𝑠 𝑔 (𝑠, 𝑋𝑠𝑢 , 𝑢 (𝑠)) 𝑑𝑠.
𝑡
Γ (𝑡, 𝑥, 𝑧) = {𝑢 ∈ 𝑈, 𝑔 (𝑡, 𝑥, 𝑢) + 𝑧𝑅 (𝑡, 𝑥, 𝑢) = 𝜓 (𝑡, 𝑥, 𝑧)} .
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Hypothesis 3. (i) 𝐴, 𝐹, and G verify Hypothesis 1.
(ii) (𝑈, U) is a measurable space. The map 𝑔 : [0, ∞)×C×
𝑚
𝑈 → 𝑅 is continuous and satisfies |𝑔(𝑡, 𝑥, 𝑢)| ≤ 𝐾𝑔 (1+|𝑥|𝐶 𝑔 )
for suitable constants 𝐾𝑔 > 0, 𝑚𝑔 > 0 and all 𝑥 ∈ C,𝑢 ∈
𝑈. The map 𝑅 : [0, ∞) × C × 𝑈 → Ξ is measurable, and
|𝑅(𝑡, 𝑠, 𝑢)| ≤ 𝐿 𝑅 for a suitable constant 𝐾𝑅 > 0 and all 𝑥 ∈
C,𝑢 ∈ 𝑈, and𝑧 ∈ Ξ∗ .
(iii) The Hamiltonian 𝜓 defined in (60) satisfies the requirements of Hypothesis 2 (with 𝐾 = 𝑅).
(iv) We fix here 𝑝 > 2, q and 𝛿 < 0 satisfying (39) with
𝜂 = 𝜂(𝑞) and such that 𝑞 > 𝑚𝑔 .
We are in a position to prove the main result of this
section.
Theorem 11. Assume that Hypothesis 3 holds, and suppose that
𝜆 verifies
∨(
(59)
𝐿2𝑧
𝐿2𝑅
)
) ∨ (−𝛿 +
2
2 (𝑝 − 1)
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𝐿2𝑅 𝑚𝑔
2 (𝑞 − 𝑚𝑔 )
− 𝜂 (𝑞) 𝑚𝑔 ) .
Let 𝜐, 𝜁 denote the function in the statement of Theorem 10.
Then, for every admissible control 𝑢 and for the corresponding
trajectory 𝑋 starting at (𝑡, 𝑥), one has
∞
𝐽 (𝑢) = 𝜐 (𝑡, 𝑥) + 𝐸 ∫ 𝑒−𝜆𝑠 [−𝜓 (𝑠, 𝑋𝑠𝑢 , 𝜁 (𝑠, 𝑋𝑠𝑢 )) + 𝜁 (𝑠, 𝑋𝑠𝑢 )
𝑡
×
+ 𝐺 (𝑠, 𝑋𝑠𝑢 ) 𝑅 (𝑠, 𝑋𝑠𝑢 , 𝑢 (𝑠)) 𝑑𝑠 + 𝐺 (𝑠, 𝑋𝑠𝑢 ) 𝑑𝑊 (𝑠) ,
𝑋𝑡𝑢 = 𝑥.
and the corresponding, possibly empty, set of minimizers
𝜆 > (−𝛿 − 𝜇 +
4. The Fundamental Relation
𝑢
𝜓 (𝑡, 𝑥, 𝑧) = inf {𝑔 (𝑡, 𝑥, 𝑢) + 𝑧𝑅 (𝑡, 𝑥, 𝑢) : 𝑢 ∈ 𝑈} .
(60)
We are now ready to formulate the assumptions we need.
Γ1 (𝑡, 𝑥) = 𝑌 (⋅, 𝑡, 𝑥) ,
Γ3 : 𝐿𝑝 (Ω, 𝑅) 󳨀→ 𝑅,
Here, 𝑔 is function on [0, ∞) × C × 𝑈 with real values. Our
purpose is to minimize the function 𝐽 over all admissible
controls.
We define in a classical way the Hamiltonian function
relative to the previous problem; for all 𝑡 ∈ [0, ∞), 𝑥 ∈
C, and 𝑧 ∈ Ξ∗ ,
𝑅 (𝑠, 𝑋𝑠𝑢 , 𝑢 (𝑠))
+ 𝑔 (𝑠, 𝑋𝑠𝑢 , 𝑢 (𝑠))] 𝑑𝑠.
(63)
Proof. Consider (58) in the probability space (Ω, F, 𝑃) with
filtration {F𝑡 }𝑡≥0 and with an {F𝑡 }𝑡≥0 -cylindrical Wiener
process {𝑊(𝑡), 𝑡 ≥ 0}. Let us define
𝑠
𝑊𝑢 (𝑠) = 𝑊 (𝑠) + ∫ 𝑅 (𝜎, 𝑋𝜎𝑢 , 𝑢 (𝜎)) 𝑑𝜎, 𝑠 ∈ [0, ∞) ,
𝑡∧𝑠
𝑇
𝜌 (𝑇) = exp (∫ −𝑅∗ (𝑠, 𝑋𝑠𝑢 , 𝑢 (𝑠)) 𝑑𝑊 (𝑠)
𝑡
−
1 𝑇 󵄨󵄨
󵄨2
∫ 󵄨𝑅 (𝑠, 𝑋𝑠𝑢 , 𝑢 (𝑠))󵄨󵄨󵄨 𝑑𝑠) .
2 𝑡 󵄨
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10
Abstract and Applied Analysis
𝑇
Let 𝑃𝑢 be the unique probability on F[0,∞) such that
= 𝐸𝑢 [exp (∫ 𝑟𝑅∗ (𝑠, 𝑋𝑠𝑢 , 𝑢 (𝑠)) 𝑑𝑊𝑢 (𝑠)
𝑡
𝑢
𝑃 |F𝑇 = 𝜌 (𝑇) 𝑃|F𝑇 .
1 𝑇 󵄨
󵄨2
− ∫ 𝑟2 󵄨󵄨󵄨𝑅 (𝑠, 𝑋𝑠𝑢 , 𝑢 (𝑠))󵄨󵄨󵄨 𝑑𝑠)
2 𝑡
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We notice that under 𝑃𝑢 , the process 𝑊𝑢 is a Wiener process.
Let us denote by {F𝑢𝑡 }𝑡≥0 the filtration generated by 𝑊𝑢 , and
completed in the usual way. Relatively to 𝑊𝑢 (58) can be
rewritten as
× exp
2
≤ 𝑒(1/2)𝑟(𝑟−1)𝑇𝐿 𝑅 𝐸𝑢
𝑑𝑋𝑢 (𝑠) = 𝐴𝑋𝑢 (𝑠) 𝑑𝑠 + 𝐹 (𝑠, 𝑋𝑠𝑢 ) 𝑑𝑠
+ 𝐺 (𝑠, 𝑋𝑠𝑢 ) 𝑑𝑊𝑢 (𝑠) ,
𝑟 (𝑟 − 1) 𝑇 󵄨󵄨
󵄨2
∫ 󵄨󵄨𝑅 (𝑠, 𝑋𝑠𝑢 , 𝑢 (𝑠))󵄨󵄨󵄨 𝑑𝑠]
2
𝑡
𝑇
𝑠 ∈ [𝑡, ∞) ,
× exp (∫ 2𝑅∗ (𝑠, 𝑋𝑠𝑢 , 𝑢 (𝑠)) 𝑑𝑊𝑢 (𝑠)
𝑡
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1 𝑇 󵄨
󵄨2
− ∫ 4󵄨󵄨󵄨𝑅 (𝑠, 𝑋𝑠𝑢 , 𝑢 (𝑠))󵄨󵄨󵄨 𝑑𝑠)
2 𝑡
𝑋𝑡𝑢 = 𝑥.
In the space (Ω, F[0,∞) , {F𝑢𝑡 }𝑡≥0 , 𝑃𝑢 ), we consider the following system of forward-backward equations:
𝑠
𝑡
+ ∫ 𝑒(𝑠−𝜎)𝐴 𝐺 (𝜎, 𝑋𝜎𝑢 ) 𝑑𝑊𝑢 (𝜎) ,
𝑇
𝑠 ∈ [𝑡, ∞) ,
𝑡
(69)
It follows that
𝑋𝑢 (𝑠) = 𝑒(𝑠−𝑡)𝐴 𝑥 (0) + ∫ 𝑒(𝑠−𝜎)𝐴 𝐹 (𝜎, 𝑋𝜎𝑢 ) 𝑑𝜎
𝑠
2
= 𝑒(1/2)𝑟(𝑟−1)𝑇𝐿 𝑅 .
−𝜆𝑠
𝐸(∫ |𝑒
𝑡
𝑢
2
1/2
𝑍 (𝑠)| 𝑑𝑠)
1/2
𝑇
󵄨2
󵄨
= 𝐸 [(∫ 󵄨󵄨󵄨󵄨𝑒−𝜆𝑠 𝑍𝑢 (𝑠)󵄨󵄨󵄨󵄨 𝑑𝑠)
𝑡
𝑢
𝜌−1 ]
1/2
𝑋𝑡𝑢
𝑇
󵄨2
󵄨
≤ (𝐸𝑢 ∫ 󵄨󵄨󵄨󵄨𝑒−𝜆𝑠 𝑍𝑢 (𝑠)󵄨󵄨󵄨󵄨 𝑑𝑠)
𝑡
= 𝑥 ∈ C,
𝑇
𝑇
𝑠
𝑠
= ∫ 𝜓 (𝜎, 𝑋𝜎𝑢 , 𝑍𝑢 (𝜎)) 𝑑𝜎,
0 ≤ 𝑠 ≤ 𝑇 < ∞.
1/2
× (𝐸𝑢 𝜌−2 )
𝑌𝑢 (𝑠) − 𝑌𝑢 (𝑇) + ∫ 𝑍𝑢 (𝜎) 𝑑𝑊𝑢 (𝜎) + 𝜆 ∫ 𝑌𝑢 (𝜎) 𝑑𝜎
𝑇
𝑠
(67)
−𝜆𝑠
𝑢
Applying the Itô formula to 𝑒 𝑌 (𝑠) and writing the backward equation in (67) with respect to the process 𝑊, we get
< ∞.
(70)
We conclude that the stochastic integral in (68) has zero
expectation. If we set 𝑠 = 𝑡 in (68) and we take expectation
with respect to 𝑃, we obtain
𝑒−𝜆𝑇 𝐸𝑌𝑢 (𝑇) − 𝑌𝑢 (𝑡)
𝑇
= 𝐸 ∫ 𝑒−𝜆𝜎 [−𝜓 (𝜎, 𝑋𝜎𝑢 , 𝑍𝑢 (𝜎))
(71)
𝑡
𝑇
+𝑍𝑢 (𝜎) 𝑅 (𝜎, 𝑋𝜎𝑢 , 𝑢 (𝜎))] 𝑑𝜎.
𝑌𝑢 (𝑠) + ∫ 𝑒−𝜆𝜎 𝑍𝑢 (𝜎) 𝑑𝑊 (𝜎)
𝑠
𝑝
By Theorem 7, 𝑌𝑢 (⋅, 𝑡, 𝑥) ∈ 𝐿 P (Ω; 𝐶𝛿 (𝑅)), so that
𝑇
= ∫ 𝑒−𝜆𝜎 [𝜓 (𝜎, 𝑋𝜎𝑢 , 𝑍𝑢 (𝜎))
𝑠
(68)
−𝑍𝑢 (𝜎) 𝑅 (𝜎, 𝑋𝜎𝑢 , 𝑢 (𝜎))] 𝑑𝜎
+ 𝑒−𝜆𝑇 𝑌𝑢 (𝑇) .
𝐸𝑢 |𝑌(𝑇, 𝑡, 𝑥)|𝑝 ≤ 𝐶 exp (−𝑝𝛿𝑇) .
(72)
By the Hölder inequality, we have that for suitable constant
𝐶 > 0,
𝐸 |𝑌 (𝑇, 𝑡, 𝑥)| = 𝐸𝑢 (𝜌−1 (𝑇) |𝑌 (𝑇, 𝑡, 𝑥)|)
Recalling that 𝑅 is bounded, we get, for all 𝑟 ≥ 1 and some
constant 𝐶,
(𝑝−1)/𝑝
1/𝑝
𝐸(|𝑌 (𝑇, 𝑡, 𝑥)|𝑝 )
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2
≤ 𝐶𝑒((𝐿 𝑅 /2(𝑝−1))−𝛿))𝑇 .
𝑇
𝐸𝑢 [𝜌(𝑇)−𝑟 ] = 𝐸𝑢 [exp 𝑟 (∫ 𝑅∗ (𝑠, 𝑋𝑠𝑢 , 𝑢 (𝑠)) 𝑑𝑊𝑢 (𝑠)
𝑡
−
≤ 𝐸(𝜌−𝑝/(𝑝−1) )
1 𝑇 󵄨󵄨
󵄨2
∫ 󵄨𝑅 (𝑠, 𝑋𝑠𝑢 , 𝑢 (𝑠))󵄨󵄨󵄨 𝑑𝑠)]
2 𝑡 󵄨
From Theorem 2, we obtain 𝐸𝑢 sup𝑠≥𝑡 𝑒𝜂𝑞𝑠 |𝑋𝑠𝑢 |𝑞 < ∞; by the
similar process, we get that
󵄨 󵄨𝑚
𝐸󵄨󵄨󵄨𝑋𝑇𝑢 󵄨󵄨󵄨 𝑔 ≤ 𝐶𝑒(𝐿 𝑅 𝑚𝑔 (2𝑞−2𝑚𝑔 )
2
−1
−𝜂(𝑞)𝑚𝑔 )𝑇
,
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Abstract and Applied Analysis
11
for suitable constant 𝐶 > 0 and
5. Existence of Optimal Control
∞
󵄨
󵄨
𝐸 ∫ 𝑒−𝜆𝜎 󵄨󵄨󵄨𝑔 (𝜎, 𝑋𝜎𝑢 , 𝑢 (𝜎))󵄨󵄨󵄨 𝑑𝜎 < ∞.
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𝑡
Since 𝑌𝑢 (𝑡, 𝑡, 𝑥) = 𝜐(𝑡, 𝑥) and 𝑍𝑢 (𝑠, 𝑡, 𝑥) = 𝜁(𝑠, 𝑋𝑠𝑢 (𝑡, 𝑥)), 𝑃a.s., for a.a. 𝑠 ∈ [𝑡, ∞), we have that
𝑒−𝜆𝑇 𝐸𝑌𝑢 (𝑇) − 𝑣 (𝑡, 𝑥)
𝑇
= 𝐸 ∫ 𝑒−𝜆𝜎 [−𝜓 (𝜎, 𝑋𝜎𝑢 , 𝜁 (𝜎, 𝑋𝜎𝑢 ))
(76)
𝑡
+𝜁 (𝜎, 𝑋𝜎𝑢 ) 𝑅 (𝜎, 𝑋𝜎𝑢 , 𝑢 (𝜎))] 𝑑𝜎.
∞
Thus, adding and subtracting 𝐸 ∫𝑡 𝑒−𝜆𝜎 𝑔(𝜎, 𝑋𝜎𝑢 , 𝑢(𝜎))𝑑𝜎 and
letting 𝑇 → ∞, we conclude that
We formulate the optimal control problem in the weak sense
following the approach of [31]. The main advantage is that we
will be able to solve the closed-loop equation in a weak sense,
and, hence, to find an optimal control, even if the feedback
law is nonsmooth.
We call (Ω, F, {F𝑡 }𝑡≥0 , 𝑃, 𝑊) an admissible setup, if
(Ω, F, {F𝑡 }𝑡≥0 , 𝑃) is a filtered probability space satisfying the
usual conditions, and 𝑊 is a cylindrical 𝑃-Wiener process
with values in Ξ, with respect to the filtration {F𝑡 }𝑡≥0 .
By an admissible control system, we mean (Ω, F,
{F𝑡 }𝑡≥0 , 𝑃, 𝑊, 𝑢, 𝑋𝑢 ), where (Ω, F, {F𝑡 }𝑡≥0 , 𝑃, 𝑊) is an
admissible setup, 𝑢 is an F𝑡 -predictable process with values
in 𝑈, and 𝑋𝑢 is a mild solution of (58). An admissible control
system will be briefly denoted by (𝑊, 𝑢, 𝑋𝑢 ) in the following.
Our purpose is to minimize the cost functional
𝐽 (𝑢) = 𝜐 (𝑡, 𝑥)
∞
∞
−𝜆𝑠
+ 𝐸∫ 𝑒
𝑡
[−𝜓 (𝑠, 𝑋𝑠𝑢 , 𝜁 (𝑠, 𝑋𝑠𝑢 ))
+
𝐽 (𝑢) = 𝐸 ∫ 𝑒−𝜆𝑠 𝑔 (𝑠, 𝑋𝑠𝑢 , 𝑢 (𝑠)) 𝑑𝑠,
𝜁 (𝑠, 𝑋𝑠𝑢 ) 𝑅
𝑡
× (𝑠, 𝑋𝑠𝑢 , 𝑢 (𝑠)) + 𝑔 (𝑠, 𝑋𝑠𝑢 , 𝑢 (𝑠))] 𝑑𝑠.
(77)
The proof is finished.
We immediately deduce the following consequences.
Theorem 12. Let 𝑡 ∈ [0, ∞) and 𝑥 ∈ C be fixed, assume
that the set-valued map Γ has nonempty values and it admits
a measurable selection Γ0 : [0, ∞) × C × Ξ∗ → 𝑈, and assume
that a control 𝑢(⋅) satisfies
𝑢 (𝑠) = Γ0 (𝑠, 𝑋𝑠𝑢 , 𝜁 (𝑠, 𝑋𝑠𝑢 )) ,
𝑃-a.s., for almost every 𝑠 ∈ [𝑡, ∞) .
𝑑𝑋 (𝑠) =𝐴𝑋 (𝑠) 𝑑𝑠 + 𝐹 (𝑠, 𝑋𝑠 ) 𝑑𝑠+𝐺 (𝑠, 𝑋𝑠 )
𝑑𝑋 (𝑠) = 𝐴𝑋 (𝑠) 𝑑𝑠 + 𝐹 (𝑠, 𝑋𝑠 ) 𝑑𝑠 + 𝐺 (𝑠, 𝑋𝑠 )
× (𝑅 (𝑠, 𝑋𝑠 , Γ0 (𝑠, 𝑋𝑠 , 𝜁 (𝑠, 𝑋𝑠 ))) 𝑑𝑠 +𝑑𝑊 (𝑠)) ,
𝑠 ∈ [𝑡, ∞) ,
𝑋𝑡 = 𝑥.
(82)
In the following sense, we say that 𝑋 is a weak solution of
(82) if there exists an admissible setup (Ω, F, {F𝑡 }𝑡≥0 , 𝑃, 𝑊)
and an F𝑡 -adapted continuous process 𝑋(𝑡) with values in 𝐻,
which solves the equation in the mild sense; namely, 𝑃-a.s.,
𝑠
𝑋 (𝑠) = 𝑒(𝑠−𝑡)𝐴 𝑥 (0) + ∫ 𝑒(𝑠−𝜎)𝐴 𝐹 (𝜎, 𝑋𝜎 ) 𝑑𝜎
×(𝑅 (𝑠, 𝑋𝑠 , Γ0 (𝑠, 𝑋𝑠 , 𝜁 (𝑠, 𝑋𝑠 ))) 𝑑𝑠 + 𝑑𝑊 (𝑠)) ,
𝑡
𝑠
+ ∫ 𝑒(𝑠−𝜎)𝐴 𝐺 (𝜎, 𝑋𝜎 ) 𝑅
𝑠 ∈ [𝑡, ∞) ,
𝑡
𝑋𝑡 = 𝑥,
(79)
𝑠
𝑡
(80)
However, under the present assumptions, we cannot guarantee
that the closed-loop equation has a solution in the mild
sense. To circumvent this difficulty, we will revert to a weak
formulation of the optimal control problem.
(83)
× (𝜎, 𝑋𝜎 , Γ0 (𝜎, 𝑋𝜎 , 𝜁 (𝜎, 𝑋𝜎 ))) 𝑑𝜎
+ ∫ 𝑒(𝑠−𝜎)𝐴 𝐺 (𝜎, 𝑋𝜎 ) 𝑑𝑊𝜎 ,
since in this case, we can define an optimal control setting
𝑢 (𝑠) = Γ0 (𝑠, 𝑋𝑠 , 𝜁 (𝑠, 𝑋𝑠 )) .
over all the admissible control system.
Our main result in this section is based on the solvability
of the closed-loop equation
(78)
Then, 𝐽(𝑡, 𝑥, 𝑢) = 𝜐(𝑡, 𝑥), and the pair (𝑢(⋅), 𝑋) is optimal for
the control problem starting from 𝑥 at time 𝑡.
Such a control can be shown to exist if there exists a solution
for the so-called closed-loop equation as follows:
(81)
𝑋𝑡 = 𝑥.
𝑠 ∈ [𝑡, ∞) ,
(84)
Theorem 13. Assume that Hypothesis 3 holds. Then, there
exists a weak solution of the closed-loop equation (82) which
is unique in law.
12
Abstract and Applied Analysis
Proof (uniqueness). Let 𝑋 be a weak solution of (82) in an
admissible setup (Ω, F, {F𝑡 }𝑡≥0 , 𝑃, 𝑊). We define
𝑇
𝜌 (𝑇) = exp (∫ −𝑅∗ (𝜎, 𝑋𝜎 , Γ0 (𝜎, 𝑋𝜎 , 𝜁 (𝜎, 𝑋𝜎 ))) 𝑑𝑊 (𝜎)
𝑡
−
1 𝑇 󵄨󵄨
󵄨2
∫ 󵄨𝑅 (𝜎, 𝑋𝜎 , Γ0 (𝜎, 𝑋𝜎 , 𝜁 (𝜎, 𝑋𝜎 )))󵄨󵄨󵄨 𝑑𝜎) .
2 𝑡 󵄨
(85)
Since 𝑅 is bounded, the Girsanov theorem ensures that there
exists a probability measure 𝑃0 such that the process
is a Wiener process (notice that 𝑅 is bounded). Then, 𝑋 is the
weak solution of (82) relatively to the probability 𝑃1 and the
Wiener process 𝑊1 .
Now, we can state the main result of this section.
Corollary 14. Assume that Hypothesis 3 holds true and 𝜆
verifies (62) Also, assume that the set-valued map Γ has
nonempty values and it admits a measurable selection Γ0 :
[0, ∞) × C × Ξ∗ → 𝑈. Then, for every 𝑡 ∈ [0, ∞) and x ∈ C
and for all admissible control system (𝑊, 𝑢, 𝑋𝑢 ), one has
𝐽 (𝑢, 𝑡, 𝑥) ≥ 𝜐 (𝑡, 𝑥) ,
𝑠
𝑊0 (𝑠) = 𝑊 (𝑠) + ∫ 𝑅 (𝜎, 𝑋𝜎 , Γ0 (𝜎, 𝑋𝜎 , 𝜁 (𝜎, 𝑋𝜎 ))) 𝑑𝜎,
𝑡∧𝑠
𝑠 ∈ [0, ∞) ,
(86)
0
is a 𝑃 -Wiener process and
𝑃0 |F𝑇 = 𝜌 (𝑇) 𝑃|F𝑇 .
(87)
Let us denote by {F0𝑡 }𝑡≥0 the filtration generated by 𝑊0 and
completed in the usual way. In (Ω, F[0,∞) , {F0𝑡 }𝑡≥0 , 𝑃0 ), 𝑋 is
a mild solution of
𝑑𝑋 (𝑠) = 𝐴𝑋 (𝑠) 𝑑𝑠 + 𝐹 (𝑡, 𝑋𝑠 ) 𝑑𝑠
+ 𝐺 (𝑠, 𝑋𝑠 ) 𝑑𝑊0 (𝑠) ,
𝑇
𝜌 (𝑇) = exp (∫ −𝑅∗ (𝜎, 𝑋𝜎 , Γ0 (𝜎, 𝑋𝜎 , 𝜁 (𝜎, 𝑋𝜎 )) 𝑑𝑊0 (𝜎)
𝑡
1 𝑇 󵄨󵄨
󵄨2
∫ 󵄨𝑅 (𝜎, 𝑋𝜎 , Γ0 (𝜎, 𝑋𝜎 , 𝜁 (𝜎, 𝑋𝜎 )))󵄨󵄨󵄨 𝑑𝜎) .
2 𝑡 󵄨
(88)
By Hypothesis 3, the joint law of 𝑋 and 𝑊0 is uniquely
determined by 𝐴, 𝐹, 𝐺, and 𝑥. Taking into account the last
displayed formula, we conclude that the joint law of 𝑋 and
𝜌(𝑇) under 𝑃0 is also uniquely determined, and consequently
so is the law of 𝑋 under 𝑃. This completes the proof of the
uniqueness part.
Proof (existence). Let (Ω, F, 𝑃) be a given complete probability space. {𝑊(𝑡), 𝑡 ≥ 0} is a cylindrical Wiener process on
(Ω, F, 𝑃) with values in Ξ, and {F𝑡 }𝑡≥0 is the natural filtration
of {𝑊(𝑡), 𝑡 ≥ 0}, augmented with the family of 𝑃-null sets. Let
𝑋(⋅) be the mild solution of
𝑑𝑋 (𝑠) = 𝐴𝑋 (𝑠) 𝑑𝑠 + 𝐹 (𝑠, 𝑋𝑠 ) 𝑑𝑠
+ 𝐺 (𝑠, 𝑋𝑠 ) 𝑑𝑊 (𝑠) ,
𝑠 ∈ [𝑡, ∞) ,
and the equality holds if
𝑢 (𝑠) = Γ0 (𝑠, 𝑋𝑠𝑢 , 𝜁 (𝑠, 𝑋𝑠𝑢 )) ,
𝑃 − 𝑎.𝑠. 𝑓𝑜𝑟 𝑎𝑙𝑚𝑜𝑠𝑡 𝑒𝑣𝑒𝑟𝑦 𝑠 ∈ [𝑡, ∞) .
Moreover, from Theorem 13, it follows that the closedloop equation (82) admits a weak solution (Ω, F,
{F𝑡 }𝑡≥0 , 𝑃, 𝑊, 𝑋) which is unique in law, and setting
𝑢 (𝑠) = Γ0 (𝑠, 𝑋𝑠 , 𝜁 (𝑠, 𝑋𝑠 )) ,
(93)
we obtain an optimal admissible control system (𝑊, 𝑢, 𝑋).
In this section, we present a simple application of the previous
results. We consider the stochastic delay partial differential
equation in the bounded domain 𝐵 ⊂ 𝑅𝑛 with smooth
boundary 𝜕𝐵 as follows:
𝑑𝑧𝑢 (𝑡, 𝜉) = Δ𝑧𝑢 (𝑡, 𝜉) 𝑑𝑡 + 𝑓 (𝑡, 𝑧𝑡𝑢 (𝜉)) 𝑑𝑡
𝑑
+ ∑ 𝑔𝑖 (𝑡, 𝑧𝑡𝑢 (𝜉)) [𝑟𝑖 (𝜉) 𝑢𝑖 (𝑡) 𝑑𝑡 + 𝑑𝑊𝑖 (𝑡)] ,
𝑖=1
𝑧0𝑢 (𝜃, 𝜉) = 𝑥 (𝜃, 𝜉) ,
𝑧𝑢 (𝑡, 𝜉) = 0,
𝜉 ∈ 𝐵, 𝜃 ∈ [−1, 0] ,
𝑡 ∈ [0, ∞) , 𝜉 ∈ 𝜕𝐵.
(94)
Here, 𝑊 = (𝑊1 , 𝑊2 , . . . , 𝑊𝑑 ) is a standard Wiener process in
𝑅𝑑 , and the functions 𝑓 : [0, +∞) × 𝐶([−1, 0], 𝑅) → 𝑅 and
𝑔𝑖 : [0, +∞) × 𝐶([−1, 0], 𝑅) → 𝑅 are Lipschitz continuous
and bounded. Setting 𝑈 as a bounded subset of 𝑅𝑑 , Ξ = 𝑅𝑑 ,
𝐻 = 𝐿2 (𝐵), and 𝑥 ∈ 𝐶([−1, 0], 𝐻). We define 𝐹 and 𝐺 as
following:
𝐹 (𝑡, 𝑥) (𝜉) = 𝑓 (𝑡, 𝑥 (𝜉)) ,
(89)
𝑑
(𝐺 (𝑡, 𝑥) 𝑧) (𝜉) = ∑ 𝑔𝑖 (𝑡, 𝑥 (𝜉)) 𝑧𝑖 (𝜉) ,
𝑋𝑡 = 𝑥,
and by the Girsanov theorem, let 𝑃 be the probability on Ω
under which
𝑠
𝑊1 (𝑠) = 𝑊 (𝑠) − ∫ 𝑅 (𝜎, 𝑋𝜎 , Γ0 (𝜎, 𝑋𝜎 , 𝜁 (𝜎, 𝑋𝜎 ))) 𝑑𝜎
(90)
(95)
𝑖=1
1
𝑡∧𝑠
(92)
6. Applications
𝑠 ∈ [𝑡, ∞) ,
𝑋𝑡 = 𝑥,
+
(91)
𝜉 ∈ 𝐵,
𝑥 ∈ 𝐶 ([−1, 0] , 𝐻) ,
𝑧 ∈ 𝐿 (Ξ, 𝐻) ,
and let 𝐴 denote the Laplace operator Δ in 𝐿2 (𝐵); with
domain 𝑊2,2 (𝐵) ⋂𝑊01,2 (𝐵) then, (94) has the form (58) and
Hypothesis 1 holds.
Abstract and Applied Analysis
13
Let us consider the optimal control problem associated
with the cost
∞
𝐽 (𝑢) = 𝐸 ∫ 𝑒−𝜆𝑡 [∫ 𝜎 (𝜉, 𝑧𝑡𝑢 (𝜉)) 𝑑𝜉 + 𝑢2 (𝑡)] 𝑑𝑡,
0
𝐵
(96)
where 𝜆 verifies (62) and 𝜎 : 𝐶([−1, 0], 𝑅) × 𝑈 → [0, ∞) is
a bounded measurable function. Define 𝑔 : 𝐶([−1, 0], 𝐻) ×
𝑈 → [0, ∞) and 𝑅 : 𝐶([−1, 0], 𝐻) × 𝑈 → Ξ by
𝑔(𝑦, 𝑢) = ∫𝐵 𝜎(𝑡, 𝑦(𝜉), 𝑢)𝑑𝜉 + 𝑢2 and 𝑅(𝑦, 𝑢) = (∫𝐵 𝑟1 (𝜉)𝑢1 𝑑𝜉,
∫𝐵 𝑟2 (𝜉)𝑢2 𝑑𝜉, . . . , ∫𝐵 𝑟𝑑 (𝜉)𝑢𝑑 𝑑𝜉) for 𝑦 ∈ 𝐶([−1, 0], 𝐻), 𝑢 =
(𝑢1 , 𝑢2 , . . . , 𝑢𝑑 ) ∈ 𝑈, respectively. It can be easily verified
that Hypothesis 3 holds true, and the set-valued map Γ has
nonempty values and it admits a measurable selection Γ0 :
[0, ∞) × C × Ξ∗ → 𝑈. Then, the closed-loop equation
(82) admits a weak solution (Ω, F, {F𝑡 }𝑡≥0 , 𝑃, 𝑊, 𝑢, 𝑧⋅ (⋅)), and
setting
𝑢 (𝑠) = Γ0 (𝑠, 𝑧𝑠 (⋅) , 𝜁 (𝑠, 𝑧𝑠 (⋅))) ,
(97)
we obtain an optimal admissible control system (𝑊, 𝑢, 𝑧(⋅)).
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Combinatorics
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Mathematical Physics
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Volume 2014
Journal of
Complex Analysis
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Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
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Stochastic Analysis
Abstract and
Applied Analysis
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International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
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Applied Mathematics
Journal of
Function Spaces
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Volume 2014
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Volume 2014
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Volume 2014
Optimization
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Volume 2014
Hindawi Publishing Corporation
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Volume 2014